Quantitative Aptitude

Algebra

3,715 Questions
All Polynomial Roots and Coefficients 2451 Zero-Product Root Finding 667 Linear System Manipulation 162 Polynomial Theorems and Reduction 85 Algebraic Identity Simplification 82 Surds and Indices 70 Modulus and Absolute Value 54 Algebraic Expression Substitution 46 Algebraic Fraction Simplification 33 Logarithmic Equations 25 Repeated Series Sum 12 Linear Equation Word Problems 9

Algebra Questions

Multiple choice

Find the roots of the following quadratic equations by using the quadratic formula $\displaystyle \frac{x - 3}{x + 3} - \frac{x + 3}{x - 3} = 6\frac{6}{7}, x \neq - 3, 3$
  1. $4,\displaystyle \frac{9}{4}$
  2. $-4,\displaystyle \frac{4}{9}$
  3. $-4,-\displaystyle \frac{4}{9}$
  4. $-4,\displaystyle \frac{9}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let y = (x-3)/(x+3). Equation: y - 1/y = 48/7. (y^2 - 1)/y = 48/7. 7y^2 - 48y - 7 = 0. (7y + 1)(y - 7) = 0. y = 7 or y = -1/7. If (x-3)/(x+3) = 7, x-3 = 7x+21, -6x = 24, x = -4. If (x-3)/(x+3) = -1/7, 7x-21 = -x-3, 8x = 18, x = 9/4.

Multiple choice

Find the roots of the following quadratic equation by using the quadratic formula $\displaystyle x + \frac{1}{x} = 3, x \neq 0$
  1. $\displaystyle \frac{3 \pm \sqrt {13}}{2}$
  2. $\displaystyle \frac{3 \pm \sqrt 5}{2}$
  3. $\displaystyle \frac{-3 + \sqrt 5}{2}$
  4. $\displaystyle \frac{-3 \pm \sqrt {13}}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given x + 1/x = 3, multiplying by x gives x^2 - 3x + 1 = 0. Using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a, we get x = (3 +/- sqrt(9 - 4)) / 2 = (3 +/- sqrt(5)) / 2.

Multiple choice

Find the roots of the following quadratic equations by using the quadratic formula $3a^2x^2 + 8abx + 4b^2 = 0, a \neq 0$
  1. $\displaystyle \frac{2b}{a}, \frac{-2b}{3a}$
  2. $\displaystyle \frac{-3b}{2a}, \frac{-2b}{3a}$
  3. $\displaystyle \frac{-2b}{a}, \frac{2b}{3a}$
  4. $\displaystyle \frac{-2b}{a}, \frac{-2b}{3a}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a, where A=3a^2, B=8ab, C=4b^2. The discriminant is (8ab)^2 - 4(3a^2)(4b^2) = 64a^2b^2 - 48a^2b^2 = 16a^2b^2. Roots are (-8ab +/- 4ab) / 6a^2, yielding -12ab/6a^2 = -2b/a and -4ab/6a^2 = -2b/3a.

Multiple choice

Find the roots of the following quadratic equation by using the quadratic formula $2x^2 - 2\sqrt 2x + 1 = 0$
  1. $\displaystyle -\frac{1}{\sqrt 2}, \frac{1}{\sqrt{2}}$
  2. $\displaystyle \frac{1}{\sqrt 2}, \frac{1}{\sqrt{2}}$.
  3. $\displaystyle -\frac{1}{ 2}, \frac{1}{{2}}$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. For 2x^2 - 2*sqrt(2)x + 1 = 0, a=2, b=-2*sqrt(2), and c=1. The discriminant is (2*sqrt(2))^2 - 4(2)(1) = 8 - 8 = 0, so the roots are equal to -b / 2a = (2*sqrt(2)) / 4 = sqrt(2) / 2 = 1 / sqrt(2).

Multiple choice

Find the discriminant of the equation and the nature of roots. Also find the roots. $2x^2 + 5 \sqrt 3x + 6 =0$
  1. $D=27$, Real and distinct, $x=\displaystyle \frac{-\sqrt 3}{2}, -2 \sqrt 3$
  2. $D=0$, Real and equal, $x=\displaystyle \frac{-\sqrt 3}{2}$
  3. $D=-2$, Imaginary, $x= \pm \displaystyle \frac{-\sqrt 3}{2}i$

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equation: 2x^2 + 5*sqrt(3)x + 6 = 0. Discriminant D = b^2 - 4ac = (5*sqrt(3))^2 - 4(2)(6) = 75 - 48 = 27. Since D > 0, roots are real and distinct. Roots = (-b +/- sqrt(D)) / 2a = (-5*sqrt(3) +/- sqrt(27)) / 4 = (-5*sqrt(3) +/- 3*sqrt(3)) / 4. Roots are -2*sqrt(3)/4 = -sqrt(3)/2 and -8*sqrt(3)/4 = -2*sqrt(3).

Multiple choice

Find the roots of the following quadratic equations by using the quadratic formula $(x^2 - 2x)^2 - 4(x^2 - 2x) + 3= 0$
  1. $-1, 3, 1\pm \sqrt{2}$
  2. $1, 3, -1\pm \sqrt{2}$
  3. $1, 3, 1\pm \sqrt{2}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let y = x^2 - 2x. Equation: y^2 - 4y + 3 = 0 -> (y-3)(y-1) = 0. Case 1: x^2 - 2x = 3 -> x^2 - 2x - 3 = 0 -> (x-3)(x+1) = 0 -> x = 3, -1. Case 2: x^2 - 2x = 1 -> x^2 - 2x - 1 = 0 -> x = (2 +/- sqrt(4 + 4)) / 2 = 1 +/- sqrt(2).

Multiple choice

Find the roots of the following quadratic equation by using the quadratic formula $(x^2 + 3x + 2)^2 - 8(x^2 + 3x) - 4 = 0$
  1. $0,1, 3, 4$,
  2. $0, 1, -3, 4$,
  3. $0, 1, -3, -4$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let u = x^2 + 3x. The equation becomes (u+2)^2 - 8u - 4 = 0. u^2 + 4u + 4 - 8u - 4 = 0. u^2 - 4u = 0. u(u-4) = 0. So x^2 + 3x = 0 or x^2 + 3x = 4. Case 1: x(x+3) = 0 -> x=0, -3. Case 2: x^2 + 3x - 4 = 0 -> (x+4)(x-1) = 0 -> x=-4, 1. Roots are 0, 1, -3, -4.

Multiple choice

Find the roots of the following quadratic equations by using the quadratic formula $\displaystyle 2 \left( \frac{x}{x + 1} \right)^2 - 5 \left( \frac{x}{x + 1} \right) + 2 = 0, x \neq - 1$
  1. $1, 2$
  2. $1, -\cfrac{1}{2}$
  3. $\cfrac{1}{2},-2$
  4. $-2, 1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let y = x / (x + 1). The equation is 2y^2 - 5y + 2 = 0. Factoring gives (2y - 1)(y - 2) = 0, so y = 1/2 or y = 2. If x/(x+1) = 1/2, then 2x = x + 1, so x = 1. If x/(x+1) = 2, then x = 2x + 2, so x = -2.

Multiple choice

Find the discriminant of the equation and the nature of roots. Also find the roots. $2x^2 - 6x + 3 = 0$
  1. $D=12$, Roots are real and distinct, $x=\displaystyle \frac{3 + \sqrt 3}{2}, \frac{3 - \sqrt 3}{2}$
  2. $D=0$, Roots are real and equal, $x=\displaystyle \frac{3 + \sqrt 3}{2}$
  3. $D=-9$, Roots are imaginary.

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For 2x^2 - 6x + 3 = 0, the discriminant D = b^2 - 4ac = (-6)^2 - 4(2)(3) = 36 - 24 = 12. Since D > 0, the roots are real and distinct, calculated via the quadratic formula as (6 +/- sqrt(12))/4 = (3 +/- sqrt(3))/2.

Multiple choice

Find the discriminant of the equation and the nature of roots. Also find the roots. $6x^2 + x - 2 = 0$
  1. $D=49$, Real and distinct roots: $\displaystyle \frac{1}{5}, \frac{-2}{3}$
  2. $D=39$, Real and distinct roots: $\displaystyle \frac{1}{2}, \frac{-2}{3}$
  3. $D=49$, Real and distinct roots: $\displaystyle \frac{1}{3}, \frac{-7}{3}$
  4. $D=49$, Real and distinct roots: $\displaystyle \frac{1}{2}, \frac{-2}{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

D = b^2 - 4ac = 1^2 - 4(6)(-2) = 1 + 48 = 49. Roots = (-b +/- sqrt(D)) / 2a = (-1 +/- 7) / 12. Roots are 6/12 = 1/2 and -8/12 = -2/3.