Algebra MCQs With Answers
Algebra questions look harder than they are once the standard identities are at your fingertips. This set of algebra MCQs with answers covers linear and simultaneous equations, quadratic equations, roots and the discriminant, identities such as (a+b)², (a−b)² and a³+b³, values of x + 1/x and its powers, laws of indices and exponential equations, logarithms, the remainder and factor theorems, and arithmetic and geometric progressions. It suits NTS, PPSC, FPSC, CSS screening and university entry test candidates who want speed on the algebra questions.
Take a scored quiz instead →Solve for x: 3(x + 5) − 2(x − 1) = 20
Expand: 3x + 15 − 2x + 2 = 20, so x + 17 = 20.
Therefore x = 3.
Solve for y: (2y + 3)/5 = (y + 6)/4
Cross-multiply: 4(2y + 3) = 5(y + 6), so 8y + 12 = 5y + 30.
3y = 18, so y = 6.
If x/2 + x/3 = 10, what is the value of x?
Using the LCM 6: 3x/6 + 2x/6 = 5x/6 = 10.
So x = 60 ÷ 5 = 12.
If 5x − 7 = 3x + 9, what is the value of x²?
5x − 3x = 9 + 7, so 2x = 16 and x = 8.
Therefore x² = 64.
The sum of three consecutive even numbers is 78. What is the largest of them?
Let the numbers be x, x + 2 and x + 4. Then 3x + 6 = 78, so x = 24.
The numbers are 24, 26 and 28, so the largest is 28.
If 2x – 5 = 15, then x is:
2x = 20 => x = 10.
Find the value of 5! (5 factorial).
5 * 4 * 3 * 2 * 1 = 120.
If x/4 = 5, then x is:
x = 5 * 4 = 20.
If 3^(2x − 1) = 243, what is the value of x?
243 = 3⁵, so 3^(2x − 1) = 3⁵.
Equating the indices: 2x − 1 = 5, so x = 3.
If 4x = 16, then x + 5 is:
x = 4, 4 + 5 = 9.
Find x if 5x – 2 = 18.
5x = 20 => x = 4.
If 7x = 49, then x – 3 is:
x = 7, 7 - 3 = 4.
If 5x = 25, then x + 2 is:
x = 5, 5 + 2 = 7.
Solve for x: 5x + 10 = 35.
5x = 25 => x = 5.
If 2x = 18, then x + 3 is:
x = 9, 9 + 3 = 12.
If 3x + 2 = 11, then x is:
3x = 9 => x = 3.
If x/5 = 4, then x is:
x = 4 * 5 = 20.
Find x if 4x = 20.
x = 20 / 4 = 5.
If 4x = 16, then x + 1 is:
x=4, 4+1=5.
Find the value of (x^3 – y^3) / (x^2 + xy + y^2).
Formula: (x-y)(x^2+xy+y^2) / (x^2+xy+y^2).
Solve 2x + 5 = x + 15.
x = 10.
If x/4 = 3, then x is:
x = 3 * 4 = 12.
If 3x – 2 = 10, then x is:
3x = 12 => x = 4.
If 4x = 32, then x + 2 is:
x=8, 8+2=10.
Find x if 5x – 5 = 20.
5x=25 => x=5.
If x/3 + 2 = 4, then x is:
x/3=2 => x=6.
If 3x = 9, then x + 5 is:
x=3, 3+5=8.
If x/2 = y/3, then x:y is:
From x/2 = y/3, cross-multiplying gives 3x = 2y, so x/y = 2/3 and x : y = 2 : 3.
Find the value of (a-b)^2 + (a+b)^2.
(a^2-2ab+b^2) + (a^2+2ab+b^2) = 2a^2+2b^2.
If log_10(100) = x, then x is:
10^2 = 100.
If 5x = 20, then x^2 is:
x=4, 4^2=16.
If 6x = 42, then x+5 is:
x=7, 7+5=12.
Find x if 3x + 9 = 27.
3x=18 => x=6.
If x/5 + 1 = 5, then x is:
x/5=4 => x=20.
If 5x = 50, then x + 10 is:
x=10, 10+10=20.
Solve for x: (x^2 – 1) / (x – 1) = 10.
(x-1)(x+1)/(x-1) = x+1 = 10 => x=9.
Find the value of a^2 – b^2.
Algebraic identity.
If x^2 = 121, x can be:
x=11 or -11.
If 3x = 1/3, then x is:
x = (1/3)/3 = 1/9.
If 9x = 81, then x-2 is:
x=9, 9-2=7.
Find x if 4x – 10 = 30.
4x=40 => x=10.
If x/5 + 2 = 6, then x is:
x/5=4 => x=20.
If 2x = 18, then x + 5 is:
x=9, 9+5=14.
Solve 3x + 12 = 30.
3x = 18 => x = 6.
If x/3 = 9, what is x?
x = 9 * 3 = 27.
If 4x – 6 = 10, then x = ?
4x = 16 => x = 4.
If x = 2, y = 3, find x^2 + y^2.
4 + 9 = 13.
If 2x + 1 = 11, then x = ?
2x = 10 => x = 5.
Find value of 2x if 4x = 20.
x=5, 2x=10.
If 3x = 12, then x + 3 is:
x=4, 4+3=7.
If x/2 = 5, then x + 2 is:
x=10, 10+2=12.
Solve for x: x/4 + x/2 = 6.
3x/4 = 6 => 3x = 24 => x = 8.
If log_2(x) = 5, what is x?
2^5 = 32.
If x+y=12 and x-y=2, what is x*y?
2x=14 x=7. y=5. 7*5=35.
If x = 1/3, what is 3x + 1?
3(1/3) + 1 = 1+1=2.
If 2x + 5 = 15, then x is:
2x=10 => x=5.
If 10x = 100, then x is:
x=10.
If 4x = 16, then x + 4 is:
x=4, 4+4=8.
Find the value of x if 2^(x+2) = 64.
2^(x+2) = 2^6 => x+2=6 => x=4.
Solve: 5x + 3 = 2x + 18.
3x = 15 => x = 5.
If 4x = 24, then x/2 is:
x=6, x/2=3.
If 3/x = 1/2, then x is:
x = 3 * 2 = 6.
If 2x + 3 = 13, then x is:
2x=10 => x=5.
If x/4 = 8, then x is:
x=32.
If a + b = 10 and ab = 21, what is the value of a² + b²?
Use a² + b² = (a + b)² − 2ab.
a² + b² = 10² − 2 × 21 = 100 − 42 = 58.
If 5x = 25, then x + 5 is:
x=5, 5+5=10.
Solve for x: 3x – 5 = 7.
3x = 12 => x = 4.
Find the value of (a+b)^2 – (a-b)^2.
(a^2+2ab+b^2) - (a^2-2ab+b^2) = 4ab.
If x^2 = 64, then x is:
x can be 8 or -8.
If 2x = 1/2, then x is:
x = (1/2)/2 = 1/4.
If 7x = 49, then x+3 is:
x=7, 7+3=10.
Find the value of x if 2x + 10 = 30.
2x=20 => x=10.
If x/2 + 5 = 10, then x is:
x/2 = 5 => x=10.
If 3x = 27, then x + 1 is:
x=9, 9+1=10.
If x = 2 and y = 3, what is 3x + 2y?
3(2) + 2(3) = 6 + 6 = 12.
If 5x – 3 = 12, then x is:
5x = 15 => x = 3.
If 2x + y = 10 and x = 3, then y is:
2(3)+y=10 => 6+y=10 => y=4.
Find x if x^2 = 81.
±9.
If y = 4x + 2, find y when x = 0.
4(0)+2 = 2.
If 6x = 36, then x = ?
x=6.
Find the value of x if 2x – 4 = 6.
2x=10 => x=5.
If x/3 = 4, then x = ?
x=12.
Find the value of x if 2^(x+3) = 32.
2^(x+3) = 2^5 => x+3=5 => x=2.
Solve for x: 5x + 10 = 3x + 20.
2x = 10 => x = 5.
If 3x=x+10, then x is:
2x=10 => x=5.
If 8x + 3y = 38 and 2x − y = 6, what is the value of y?
From the second equation, y = 2x − 6. Substituting: 8x + 3(2x − 6) = 38, so 14x = 56 and x = 4.
Then y = 2(4) − 6 = 2.
If 2x + 4 = 12, then x is:
2x=8 => x=4.
If x = 1, y = 2, find x + 2y.
1 + 2(2) = 5.
Find value of 2x if 4x = 8.
x=2, 2x=4.
Find the value of x: 3x – 5 = x + 7.
2x = 12 => x = 6.
If x+y=10 and x-y=4, what is xy?
2x=14 x=7. y=3. 7*3=21.
If log_x(64) = 3, then x is:
x^3 = 64.
If 4x = 20, then x + 5 is:
x=5, 5+5=10.
A number is increased by 5 and the result is multiplied by 3, giving 51. What is the number?
Let the number be x. Then 3(x + 5) = 51, so x + 5 = 17.
Therefore x = 12.
Find the value of (x^2 – y^2) / (x – y) if x=10, y=5.
(100-25)/(10-5) = 75/5 = 15.
If 2x + 3y = 12 and x – y = 1, find x and y.
x=y+1. 2(y+1)+3y=12 => 5y+2=12 => 5y=10 => y=2, x=3.
If 3x = 12, then 5x + 2 = ?
x=4. 5(4)+2 = 22.
If x/2 + x/3 = 10, find x.
3x/6 + 2x/6 = 10 => 5x/6=10 => x=12.
Find the value of log_10(1000).
10^3=1000.
If 7x = 49, then x is:
x=7.
If x = 5, find 2x + 10.
10+10=20.
If 2x + 3 = 11, what is the value of 3x − 2?
2x = 11 − 3 = 8, so x = 4.
Then 3x − 2 = 12 − 2 = 10.
Solve for x: x^2 – 16 = 0.
x^2 = 16 => x = ±4.
If x/3 = 10, then x is:
x = 10 * 3 = 30.
If 3x + 5 = 20, find x.
3x = 15 => x = 5.
If log x = 2 (base 10), then x is:
x = 10^2 = 100.
If y = 2x + 1, find y when x = 3.
2(3)+1 = 7.
Solve for x: 3^x * 3^2 = 3^6.
3^(x+2) = 3^6 => x+2=6 => x=4.
If x + y = 10 and x – y = 2, what is x*y?
2x = 12 => x=6. y=4. xy = 24.
Value of log_2(16)?
2^4 = 16.
2^(x+2) = 32. x=?
2^(x+2) = 2^5 => x+2=5 => x=3.
Solve 5x-10=0.
5x=10 => x=2.
Simplify x^6 / x^2.
6-2=4.
A father is now four times as old as his son. After 16 years he will be twice as old as his son. What is the son’s present age?
Let the son's age be x, so the father is 4x. After 16 years: 4x + 16 = 2(x + 16).
4x + 16 = 2x + 32, so 2x = 16 and x = 8 years (father 32).
If x^2 – 7x + 10 = 0, find the roots.
(x-2)(x-5) = 0 => x=2, 5.
Find the value of log_3(81).
3^4 = 81.
If 2^(x+1) = 16, find x.
- Write 16 as a power of 2: 16 = 2⁴, so the equation becomes 2^(x+1) = 2⁴.
- When the bases are the same, the exponents must be equal, so x + 1 = 4 and x = 3.
- Check by substituting back: 2^(3+1) = 2⁴ = 16, which matches the right side.
- Option C 4 is the usual trap, because 4 is the power of 2 that gives 16, but the exponent here is x + 1, not x.
- The same method solves any index equation of the form a^(x+k) = a^n: match the bases first, then equate the exponents.
Solve 2x – 3 = 7.
2x = 10 => x = 5.
Simplify: (x^2 * x^4) / x^3.
x^(2+4-3) = x^3.
Solve 3x + 2 = 14.
3x = 12 => x = 4.
Solve: x^2 – 5x + 6 = 0.
(x-2)(x-3) = 0 => x = 2, 3.
If (x+y)^2 = 100 and (x-y)^2 = 16, find xy.
4xy = (x+y)^2 - (x-y)^2 = 100 - 16 = 84 => xy = 21.
Solve 3^(x-1) = 9.
3^(x-1) = 3^2 => x-1=2 => x=3.
Remainder of (x^3 – 1) / (x-1)?
1^3 - 1 = 0.
Simplify x^3 / x^2?
x^(3-2)=x.
Solve 2x+1=9?
2x=8 => x=4.
Factorize x^2 – 9.
Difference of squares: a^2 - b^2 = (a-b)(a+b). x^2 - 3^2 = (x-3)(x+3).
Solve for x in 5^x = 125.
5^x = 5^3 => x = 3.
Evaluate log_2(8).
2^3 = 8.
Remainder of (x^2 + 1) / (x-1)?
P(1) = 1^2 + 1 = 2.
Simplify x^2 * x^3.
x^(2+3) = x^5.
If x + 1/x = 4, find x^2 + 1/x^2.
x^2 + 1/x^2 = (x + 1/x)^2 - 2 = 16 - 2 = 14.
Simplify: (x^3 – y^3) / (x – y).
(x^3 - y^3) = (x-y)(x^2 + xy + y^2). Division gives x^2 + xy + y^2.
Find the remainder when x^2 + 2x + 1 is divided by (x+1).
P(-1) = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0.
Simplify x^0.
Any number^0 = 1.
Solve for x: 3x – 5 = 10.
3x = 15 => x = 5.
If x + y = 10 and xy = 21, find the value of x^3 + y^3.
x^3 + y^3 = (x+y)(x^2-xy+y^2) = (x+y)((x+y)^2-3xy) = 10(100-3*21) = 10(100-63) = 10*37 = 370.
If x – 1/x = 3, find x^2 + 1/x^2.
x^2 + 1/x^2 = (x - 1/x)^2 + 2 = 3^2 + 2 = 11.
Remainder of (x^2 – 5x + 6) / (x – 2)?
- By the remainder theorem, dividing by (x − 2) leaves a remainder equal to P(2).
- Work it out: P(2) = 2² − 5(2) + 6 = 4 − 10 + 6 = 0.
- A remainder of 0 means (x − 2) divides the expression exactly, so it is a factor.
- Factorising confirms it: x² − 5x + 6 = (x − 2)(x − 3).
- Change the divisor to (x − 1) and the remainder becomes P(1) = 1 − 5 + 6 = 2, so the divisor decides the value.
Simplify x^a / x^b.
- When dividing powers that share the same base, subtract the indices: x^a ÷ x^b = x^(a−b).
- Test it with numbers: 2⁵ ÷ 2³ = 32 ÷ 8 = 4, and 2^(5−3) = 2² = 4.
- Option B x^(a+b) is the rule for multiplying powers, not for dividing them.
- If a equals b the result becomes x⁰ = 1, so option D is correct only in that special case.
- When a is smaller than b the index turns negative, and x^(−n) simply means 1/xⁿ.
If 2x − 3y = 4 and x + y = 7, what is the value of x?
From x + y = 7, y = 7 − x. Substituting: 2x − 3(7 − x) = 4, so 5x − 21 = 4.
5x = 25, so x = 5 (and y = 2).
If 4x + 3y = 18 and xy = 6, find the value of 16x^2 + 9y^2.
Squaring both sides: (4x + 3y)^2 = 18^2 => 16x^2 + 24xy + 9y^2 = 324 => 16x^2 + 24(6) + 9y^2 = 324 => 16x^2 + 9y^2 = 324 - 144 = 180.
If x + 1/x = 6, find the value of x^2 + 1/x^2.
x^2 + 1/x^2 = (x + 1/x)^2 - 2 = 6^2 - 2 = 36 - 2 = 34.
Solve for x: log_10(x) + log_10(x – 3) = 1.
log_10[x(x - 3)] = 1 => x(x - 3) = 10^1 = 10 => x^2 - 3x - 10 = 0 => (x - 5)(x + 2) = 0. Since log requires positive input, x = 5.
Find the remainder when 2x^3 – 5x^2 + 3x + 7 is divided by (x – 1).
By Remainder Theorem, evaluate P(1): 2(1)^3 - 5(1)^2 + 3(1) + 7 = 2 - 5 + 3 + 7 = 7.
Simplify: (x^(a-b))^(a+b) / (x^(a+b))^(a-b).
Numerator = x^(a^2 - b^2). Denominator = x^(a^2 - b^2). Division gives x^(a^2 - b^2 - (a^2 - b^2)) = x^0 = 1.
How many terms are there in the sequence 5, 10, 15, …, 200?
Arithmetic sequence with a = 5, d = 5, l = 200. l = a + (n-1)d => 200 = 5 + (n-1)*5 => 195 = 5(n-1) => n - 1 = 39 => n = 40.
Solve for x: sqrt(4x + 9) = 7.
4x + 9 = 7^2 = 49 => 4x = 40 => x = 10.
If 3x + 2y = 13 and xy = 6, find the value of 9x^2 + 4y^2.
Squaring both sides of (3x + 2y): (3x + 2y)^2 = 9x^2 + 12xy + 4y^2. 13^2 = 9x^2 + 12(6) + 4y^2 => 169 = 9x^2 + 72 + 4y^2 => 9x^2 + 4y^2 = 169 - 72 = 97.
If x – 1/x = 4, find the value of x^3 – 1/x^3.
x^3 - 1/x^3 = (x - 1/x)^3 + 3(x - 1/x) = 4^3 + 3(4) = 64 + 12 = 76.
Solve for x: log_2(x + 3) + log_2(x – 3) = 4.
log_2[(x + 3)(x - 3)] = 4 => x^2 - 9 = 2^4 = 16 => x^2 = 25 => x = 5 (since log requires x > 3).
Find the remainder when x^3 – 3x^2 + 4x – 5 is divided by (x – 2).
By Remainder Theorem, substitute x = 2: P(2) = 2^3 - 3(2^2) + 4(2) - 5 = 8 - 12 + 8 - 5 = -1.
Simplify: (x^b / x^c)^(b+c) * (x^c / x^a)^(c+a) * (x^a / x^b)^(a+b).
Expression = x^((b-c)(b+c)) * x^((c-a)(c+a)) * x^((a-b)(a+b)) = x^(b^2-c^2 + c^2-a^2 + a^2-b^2) = x^0 = 1.
Solve for x: sqrt(5x – 4) = 6.
- Square both sides to clear the root: 5x − 4 = 6² = 36.
- Solve what is left: 5x = 40, so x = 8.
- Substitute to verify: √(5 × 8 − 4) = √36 = 6, which matches the question.
- Always test your answer in the original equation, because squaring can create values that do not really work.
- The expression under a square root cannot be negative, so 5x − 4 ≥ 0 means x must be at least 0.8.
If 2x – 3y = 7 and xy = 3, find the value of 4x^2 + 9y^2.
(2x - 3y)^2 = 4x^2 - 12xy + 9y^2 => 7^2 = 4x^2 - 12(3) + 9y^2 => 49 = 4x^2 - 36 + 9y^2 => 4x^2 + 9y^2 = 49 + 36 = 85.
If x^2 + 1/x^2 = 27, find the positive value of x – 1/x.
(x - 1/x)^2 = x^2 + 1/x^2 - 2 = 27 - 2 = 25 => x - 1/x = sqrt(25) = 5.
Find the value of x if log_3(x) + log_3(x – 8) = 2.
log_3[x(x - 8)] = 2 => x(x - 8) = 3^2 = 9 => x^2 - 8x - 9 = 0 => (x - 9)(x + 1) = 0. Since log requires positive argument, x = 9.
Find the value of k if (x – 2) is a factor of polynomial x^3 – 4x^2 + kx – 8.
By Factor Theorem, P(2) = 0 => 2^3 - 4(2^2) + k(2) - 8 = 0 => 8 - 16 + 2k - 8 = 0 => 2k - 16 = 0 => k = 8.
Find the value of (x^(a-b))^(a+b) * (x^(b-c))^(b+c) * (x^(c-a))^(c+a).
Exponents simplify using (p-q)(p+q) = p^2 - q^2: x^(a^2 - b^2) * x^(b^2 - c^2) * x^(c^2 - a^2) = x^(a^2 - b^2 + b^2 - c^2 + c^2 - a^2) = x^0 = 1.
How many terms are there in the AP: 7, 13, 19, …, 205?
a = 7, d = 6, l = 205. l = a + (n-1)d => 205 = 7 + (n-1)*6 => 198 = (n-1)*6 => n-1 = 33 => n = 34.
Find the value of x: sqrt(3x + 1) = 7.
3x + 1 = 7^2 = 49 => 3x = 48 => x = 16.
If x : y = 3 : 4, find the value of (5x – 2y) : (7x + 2y).
Substitute x = 3 and y = 4: (5*3 - 2*4) : (7*3 + 2*4) = (15 - 8) : (21 + 8) = 7 : 29.
If log_10(2) = 0.3010, what is the value of log_10(80)?
log_10(80) = log_10(8 * 10) = log_10(2^3) + log_10(10) = 3 * log_10(2) + 1 = 3(0.3010) + 1 = 0.9030 + 1 = 1.9030.
If x + 1/x = 5, find the value of x^3 + 1/x^3.
x^3 + 1/x^3 = (x + 1/x)^3 - 3(x + 1/x) = 5^3 - 3(5) = 125 - 15 = 110.
Solve for x: 3^(2x – 1) = 27^(x – 2).
27 = 3^3 => 3^(2x - 1) = (3^3)^(x - 2) = 3^(3x - 6). Bases equal => 2x - 1 = 3x - 6 => 3x - 2x = 6 - 1 => x = 5.
If x – y = 6 and x^2 + y^2 = 116, find the value of xy.
(x - y)^2 = x^2 + y^2 - 2xy => 6^2 = 116 - 2xy => 36 = 116 - 2xy => 2xy = 80 => xy = 40.
If log_2(x – 1) = 3, find the value of x.
log_2(x - 1) = 3 => x - 1 = 2^3 => x - 1 = 8 => x = 9.
Solve for x: (2x + 5) / 3 = (x + 7) / 2.
Cross-multiply: 2(2x + 5) = 3(x + 7) => 4x + 10 = 3x + 21 => 4x - 3x = 21 - 10 => x = 11.
If x + y = 12 and xy = 35, find the value of x^3 + y^3.
x^3 + y^3 = (x + y)^3 - 3xy(x + y) = 12^3 - 3(35)(12) = 1728 - 1260 = 468.
If x^4 + 1/x^4 = 47, find the positive value of x + 1/x.
(x^2 + 1/x^2)^2 = x^4 + 1/x^4 + 2 = 47 + 2 = 49 => x^2 + 1/x^2 = 7. Then (x + 1/x)^2 = 7 + 2 = 9 => x + 1/x = 3.
If 2^(x – 1) + 2^(x + 1) = 320, find x.
Factor out 2^(x-1): 2^(x-1) [1 + 2^2] = 320 => 5 * 2^(x-1) = 320 => 2^(x-1) = 64 = 2^6 => x - 1 = 6 => x = 7.
Find the value of k for which the quadratic equation 2x^2 + kx + 3 = 0 has real and equal roots.
Equal roots => D = b^2 - 4ac = 0 => k^2 - 4(2)(3) = 0 => k^2 = 24 => k = ±sqrt(24) = ±2*sqrt(6).
If x^2 – 3x + 1 = 0, find the value of x^2 + 1/x^2.
Divide equation by x: x - 3 + 1/x = 0 => x + 1/x = 3. Squaring both sides: (x + 1/x)^2 = 3^2 => x^2 + 1/x^2 + 2 = 9 => x^2 + 1/x^2 = 7.
Find the value of x if 3^(x + 3) – 3^x = 234.
Factor out 3^x: 3^x (3^3 - 1) = 234 => 3^x (27 - 1) = 234 => 26 * 3^x = 234 => 3^x = 9 = 3^2 => x = 2.
If log_10(x) + log_10(5) = 2, find the value of x.
- Logs are added by multiplying the numbers inside them: log₁₀(x) + log₁₀(5) = log₁₀(5x).
- So log₁₀(5x) = 2, which gives 5x = 10² = 100 and therefore x = 20.
- Verify it: log₁₀(20) + log₁₀(5) = log₁₀(100) = 2.
- Option D 50 is wrong because log₁₀(250) is about 2.4, not 2.
- The matching rule for subtraction is log(a) − log(b) = log(a/b), which handles the reverse form of this question.
If x – 1/x = 4, find the value of x^2 + 1/x^2.
Squaring both sides: (x - 1/x)^2 = 4^2 => x^2 + 1/x^2 - 2 = 16 => x^2 + 1/x^2 = 18.
If 2x + 3y = 12 and xy = 6, find the value of 4x^2 + 9y^2.
(2x + 3y)^2 = 4x^2 + 9y^2 + 12xy => 12^2 = 4x^2 + 9y^2 + 12(6) => 144 = 4x^2 + 9y^2 + 72 => 4x^2 + 9y^2 = 72.
If a + b + c = 9 and ab + bc + ca = 26, find the value of a^2 + b^2 + c^2.
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) => 9^2 = a^2 + b^2 + c^2 + 2(26) => 81 = a^2 + b^2 + c^2 + 52 => a^2 + b^2 + c^2 = 29.
If 3^(x + y) = 81 and 81^(x – y) = 3, find the value of x.
3^(x+y) = 3^4 => x + y = 4. (3^4)^(x-y) = 3^1 => 4(x - y) = 1 => x - y = 1/4. Adding both equations: 2x = 4 + 1/4 = 17/4 => x = 17/8.
If the roots of the equation x^2 – kx + 16 = 0 are equal, find the value of k.
For equal roots, discriminant D = b^2 - 4ac = 0 => (-k)^2 - 4(1)(16) = 0 => k^2 - 64 = 0 => k = ±8.
If x^2 + 1/x^2 = 23, find the positive value of x + 1/x.
(x + 1/x)^2 = x^2 + 1/x^2 + 2 = 23 + 2 = 25 => x + 1/x = sqrt(25) = 5.
Find the value of x if 4^(x + 1) + 4^x = 320.
Factor out 4^x: 4^x (4 + 1) = 320 => 5 * 4^x = 320 => 4^x = 64 => 4^x = 4^3 => x = 3.
Find the value of x if 3^(2x – 1) = 27.
27 = 3^3. Equating exponents: 3^(2x - 1) = 3^3 => 2x - 1 = 3 => 2x = 4 => x = 2.
If (a + b) = 7 and ab = 12, what is the value of (a^2 + b^2)?
(a + b)^2 = a^2 + b^2 + 2ab => 7^2 = a^2 + b^2 + 2(12) => 49 = a^2 + b^2 + 24 => a^2 + b^2 = 25.
How many terms are there in the arithmetic progression: 5, 9, 13, …, 81?
a = 5, d = 4, nth term T_n = 81. T_n = a + (n - 1)d => 81 = 5 + (n - 1)4 => 76 = 4(n - 1) => n - 1 = 19 => n = 20.
Find the value of log_2(64).
64 = 2^6. Therefore log_2(64) = log_2(2^6) = 6.
If 4x + 3y = 18 and 2x – y = 4, find the value of x.
From second equation: y = 2x - 4. Substitute into first: 4x + 3(2x - 4) = 18 => 4x + 6x - 12 = 18 => 10x = 30 => x = 3.
Find the value of (102)^2 using algebraic identity.
Use (a + b)^2 = a^2 + 2ab + b^2 where a=100, b=2: 100^2 + 2(100)(2) + 2^2 = 10000 + 400 + 4 = 10404.
Find the value of x if 2^(x+3) = 128.
128 = 2^7. Equating exponents: x + 3 = 7 => x = 4.
If log_3(x) = 4, what is the value of x?
log_3(x) = 4 => x = 3^4 = 81.
Solve for x: (x + 3) / 2 = (2x – 1) / 3.
Cross-multiply: 3(x + 3) = 2(2x - 1) => 3x + 9 = 4x - 2 => 4x - 3x = 9 + 2 => x = 11.
If 3x + 4y = 25 and x = 3, what is the value of y?
3(3) + 4y = 25 => 9 + 4y = 25 => 4y = 16 => y = 4.
If x^2 + y^2 = 25 and xy = 12, what is (x + y)^2?
(x + y)^2 = x^2 + y^2 + 2xy = 25 + 2(12) = 25 + 24 = 49.
If 5^(x-1) = 125, what is the value of x?
125 = 5^3. So 5^(x-1) = 5^3 => x - 1 = 3 => x = 4.
Find the value of (x – 2) if x^2 – 4x + 4 = 0.
x^2 - 4x + 4 = (x - 2)^2 = 0 => x - 2 = 0.
Solve for x: 5x – 7 = 2x + 8.
Bring variables to one side: 5x - 2x = 8 + 7 => 3x = 15 => x = 5.
If x/y = 3/4, what is (2x + y) / (x + 2y)?
Divide numerator and denominator by y: (2(x/y) + 1) / ((x/y) + 2) = (2(3/4) + 1) / (3/4 + 2) = (3/2 + 1) / (11/4) = (5/2) / (11/4) = (5/2) * (4/11) = 10/11.
What is the sum of the roots of the quadratic equation x^2 – 7x + 12 = 0?
For ax^2 + bx + c = 0, sum of roots = -b/a. Here a=1, b=-7, c=12. Sum of roots = -(-7)/1 = 7.
If log_10(x) = 3, what is the value of x?
log_10(x) = 3 => x = 10^3 = 1000.
Solve for x: x^2 – 9 = 0.
x^2 = 9 => x = sqrt(9) = ±3.
Find the value of x if (x/3) + (x/4) = 7.
Taking LCM of 3 and 4 (which is 12): (4x + 3x)/12 = 7 => 7x/12 = 7 => 7x = 84 => x = 12.
If x^2 – y^2 = 36 and x – y = 4, what is the value of x + y?
Algebraic identity: x^2 - y^2 = (x - y)(x + y). Substituting values: 36 = 4 * (x + y) => x + y = 36 / 4 = 9.
If 3^(x-1) = 27, what is the value of x?
27 = 3^3. So 3^(x-1) = 3^3 => x - 1 = 3 => x = 4.
If 2x + 3y = 12 and x = 3, what is the value of y?
Substitute x = 3: 2(3) + 3y = 12 => 6 + 3y = 12 => 3y = 6 => y = 2.
If 4^x = 64, what is 2^(2x-1)?
4^x = 4^3 => x = 3. Now 2^(2(3)-1) = 2^(6-1) = 2^5 = 32.
If x + 1/x = 5, what is the value of x^2 + 1/x^2?
Squaring both sides: (x + 1/x)^2 = 5^2 => x^2 + 2 + 1/x^2 = 25 => x^2 + 1/x^2 = 23.
If x – y = 6 and x * y = 16, what is the value of x^2 + y^2?
Use (x − y)² = x² + y² − 2xy. Substituting the values: 6² = x² + y² − 2(16), so 36 = x² + y² − 32 and x² + y² = 68.
If 3^x = 81, what is the value of 2^(2x – 3)?
3^x = 3^4 => x = 4. Then 2^(2(4) - 3) = 2^(8 - 3) = 2^5 = 32.
Solve for x: 5(x – 3) = 2(x + 6).
5x - 15 = 2x + 12 => 3x = 27 => x = 9.
If 6^x = 216, what is x?
216 = 6^3, so x = 3.
Simplify: (x^2 – 9) / (x – 3).
x^2 - 9 = (x - 3)(x + 3). Dividing by (x - 3) gives (x + 3).
What is the value of x if 2^(x+1) = 32?
32 = 2^5, so x + 1 = 5 => x = 4.
Solve for x in the quadratic equation x^2 – 7x + 12 = 0.
Factorize the quadratic equation by finding two numbers that multiply to 12 and add to -7: (x − 3)(x − 4) = 0. Setting each factor to zero gives x = 3 and x = 4.
Solve for x: 5x – 3 = 2x + 12
Subtract 2x from both sides: 3x - 3 = 12. Add 3 to both sides: 3x = 15 => x = 5.
What is the expansion of (x + y)^3?
By binomial expansion identity, (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3.
If 3^x = 81, what is x?
Express 81 as a power of 3: 81 = 3^4. Therefore, 3^x = 3^4 => x = 4.
Find the discriminant of the quadratic equation 2x^2 – 4x + 3 = 0.
Discriminant D = b^2 - 4ac. Here a=2, b=-4, c=3. D = (-4)^2 - 4(2)(3) = 16 - 24 = -8.
If 2^(2x) = 16, what is x?
2^(2x) = 2^4 => 2x = 4 => x = 2.
If x/4 – 2 = 3, what is x?
x/4 = 3 + 2 => x/4 = 5 => x = 5 * 4 = 20.
Find the sum of the first 20 terms of the arithmetic progression: 3, 7, 11, 15, …
Formula Sn = (n/2)[2a + (n-1)d]. Here a = 3, d = 4, n = 20. S20 = (20/2)[2(3) + 19(4)] = 10 * [6 + 76] = 10 * 82 = 820.
What is the 5th term of the geometric progression: 2, 6, 18, 54, …?
First term a = 2, common ratio r = 3. 5th term = a * r^4 = 2 * (3^4) = 2 * 81 = 162.
Find the sum of an infinite geometric series with first term 10 and common ratio 0.5.
S_inf = a / (1 - r) = 10 / (1 - 0.5) = 10 / 0.5 = 20.
Solve for x: 3x + 7 = 22
Subtract 7 from both sides: 3x = 22 - 7 => 3x = 15. Divide by 3: x = 15 / 3 = 5.
Simplify the expression: (2x^3)^3
Applying power rules (a*b)^n = a^n * b^n and (x^m)^n = x^(m*n): (2x^3)^3 = 2^3 * x^(3*3) = 8x^9.
If x/3 + 4 = 9, what is the value of x?
Subtract 4 from both sides: x/3 = 5. Multiply both sides by 3: x = 15.
If a – b = 5 and a + b = 15, find the value of a^2 – b^2.
Use difference of squares identity: a^2 - b^2 = (a - b)(a + b) = 5 * 15 = 75.
What is the 10th term of the Arithmetic Progression: 2, 5, 8, 11, …?
In AP, a10 = a + (10 - 1)d = 2 + 9 * 3 = 2 + 27 = 29.
What is the common ratio of the geometric progression: 3, 6, 12, 24, …?
Common ratio r = Term2 / Term1 = 6 / 3 = 2.
What is the sum of the series 1 + 2 + 3 + … + 100?
Sum = n * (n + 1) / 2 = 100 * 101 / 2 = 50 * 101 = 5050.