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

The roots of the quadratic equation $\displaystyle \sqrt 2 x^2 + 7x + 5\sqrt 2 = 0$ are
  1. The two roots are $\displaystyle \frac{3}{\sqrt{2}} $ and $ -\dfrac{6}{\sqrt 5}$
  2. The two roots are $\displaystyle \sqrt 7 $ and $ -\dfrac{5}{\sqrt 2}$
  3. The two roots are $\displaystyle -\frac{7}{\sqrt{2}}  $ and $ -\dfrac{6}{\sqrt 5}$
  4. The two roots are $\displaystyle -\sqrt 2 $ and $ -\dfrac{5}{\sqrt 2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solve sqrt(2)x^2 + 7x + 5*sqrt(2) = 0. Using the quadratic formula or factoring: sqrt(2)x^2 + 2x + 5x + 5*sqrt(2) = 0. sqrt(2)x(x + sqrt(2)) + 5(x + sqrt(2)) = 0. (sqrt(2)x + 5)(x + sqrt(2)) = 0. Roots are -sqrt(2) and -5/sqrt(2).

Multiple choice

The zeros of the quadratic polynomial $x^2 + kx + k, k \neq 0$

  1. cannot both be positive

  2. cannot both be negative

  3. are always unequal

  4. are always equal
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For x^2 + kx + k = 0, product of roots = k and sum of roots = -k. If roots are positive, product must be positive (k > 0) and sum must be positive (-k > 0 => k < 0). This is a contradiction. Thus, they cannot both be positive.

Multiple choice

Find the roots of the following equation. $\displaystyle \frac{1}{x +4} - \frac{1}{x - 7} = \frac{11}{30}, x \neq {-4 , 7}$
  1. $-1,-2$
  2. $-1,-3$
  3. $1,3$
  4. $1,2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Simplifying the left side of the equation gives -11 / (x^2 - 3x - 28) = 11 / 30. Dividing both sides by 11 and cross-multiplying yields x^2 - 3x - 28 = -30, which simplifies to the quadratic equation x^2 - 3x + 2 = 0. Factoring this equation gives (x - 1)(x - 2) = 0, so the roots are 1 and 2.

Multiple choice

The roots of the quadratic equation $\displaystyle 2x^2 - x + \frac{1}{8} = 0$ are
  1.  The roots are $\displaystyle \frac{2}{3}, \frac{3}{4}$.
  2.  The roots are $\displaystyle \frac{1}{4}, \frac{1}{4}$.
  3.  The roots are $\displaystyle \frac{1}{2}, \frac{3}{2}$.
  4.  The roots are $\displaystyle \frac{5}{2}, \frac{2}{7}$.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation 2x^2 - x + 1/8 = 0 can be multiplied by 8 to get 16x^2 - 8x + 1 = 0. This factors as (4x - 1)^2 = 0, which gives x = 1/4 as a repeated root.

Multiple choice

Find the roots of the quadratic equation by applying the quadratic formula $\displaystyle 2x^2 + x - 4 = 0$
  1. $\displaystyle \frac{1 \pm \sqrt {33}}{4}$
  2. $\displaystyle \frac{-1 \pm \sqrt {33}}{4}$
  3. $\displaystyle \frac{1 \pm \sqrt {31}}{4}$
  4. $\displaystyle \frac{-1 \pm \sqrt {31}}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For 2x^2 + x - 4 = 0, a=2, b=1, c=-4. The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. x = (-1 +/- sqrt(1^2 - 4*2*(-4))) / (2*2) = (-1 +/- sqrt(1 + 32)) / 4 = (-1 +/- sqrt(33)) / 4.

Multiple choice

Find the value of $p$ for which the quadratic equation $x^2 + p(4x + p - 1) + 2 = 0$ has equal roots ?
  1. $\displaystyle -1, \frac{2}{3}$
  2. $3 , 5$
  3. $\displaystyle -1, \frac{4}{3}$
  4. $\displaystyle \frac{3}{4}, 2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equation: x^2 + 4px + p^2 - p + 2 = 0. For equal roots, discriminant D = (4p)^2 - 4(1)(p^2 - p + 2) = 0. 16p^2 - 4p^2 + 4p - 8 = 0. 12p^2 + 4p - 8 = 0. 3p^2 + p - 2 = 0. (3p-2)(p+1) = 0. Roots are p = 2/3, p = -1.