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 equation $x\,(x\,-\,1)\,=\,1$
  1. $\displaystyle\,x\,=\,\frac{1\,\pm\,\sqrt5}{2}$
  2. $\displaystyle\,x\,=\,\frac{2\,\pm\,\sqrt5}{2}$
  3. $\displaystyle\,x\,=\,\frac{3\,\pm\,\sqrt5}{2}$
  4. $\displaystyle\,x\,=\,\frac{4\,\pm\,\sqrt5}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

x^2 - x - 1 = 0. Using quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / 2a = (1 +/- sqrt(1 - 4(1)(-1))) / 2 = (1 +/- sqrt(5)) / 2.

Multiple choice

Find the roots of the following quadratic equation $4p^2\,+\,7\,=\,12p$
  1. $\displaystyle\,p\,=\,\frac{5\,\pm \,\sqrt2}{2}$
  2. $\displaystyle\,p\,=\,\frac{4\,\pm \,\sqrt2}{2}$
  3. $\displaystyle\,p\,=\,\frac{3\,\pm \,\sqrt2}{2}$
  4. $\displaystyle\,p\,=\,\frac{2\,\pm \,\sqrt2}{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

4p^2 - 12p + 7 = 0. Using quadratic formula p = [-b +/- sqrt(b^2 - 4ac)] / 2a. p = [12 +/- sqrt(144 - 112)] / 8 = [12 +/- sqrt(32)] / 8 = [12 +/- 4*sqrt(2)] / 8 = [3 +/- sqrt(2)] / 2.

Multiple choice

Find the roots of following quadratic equation $5m^2\,-\,2m\,=\,2$
  1. $\displaystyle\,m\,=\,\frac{1\,\pm\,\sqrt{9}}{5}$
  2. $\displaystyle\,m\,=\,\frac{-1\,\pm\,\sqrt{11}}{5}$
  3. $\displaystyle\,m\,=\,\frac{1\,\pm\,\sqrt{11}}{5}$
  4. $\displaystyle\,m\,=\,\frac{1\,\pm\,\sqrt{13}}{5}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

5m^2 - 2m - 2 = 0. Using the quadratic formula m = [-b +/- sqrt(b^2 - 4ac)] / 2a. m = [2 +/- sqrt(4 - 4*5*(-2))] / 10 = [2 +/- sqrt(4 + 40)] / 10 = [2 +/- sqrt(44)] / 10 = [2 +/- 2*sqrt(11)] / 10 = [1 +/- sqrt(11)] / 5.

Multiple choice

Find the roots of following quadratic equation $x^2\,+\,3x\,-\,2\,=\,0$
  1. $\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{15}}{2}$
  2. $\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{17}}{2}$
  3. $\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{19}}{2}$
  4. $\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{21}}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a. Here a=1, b=3, c=-2. x = (-3 +/- sqrt(9 - 4*1*-2)) / 2 = (-3 +/- sqrt(17)) / 2.

Multiple choice

Roots of quadratic equation $7p^2\,-\,5p\,-\,2\,=\,0$ are
  1. $\displaystyle\,p\,=\,3,\,\frac{-2}{7}$
  2. $\displaystyle\,p\,=\,2,\,\frac{-2}{7}$
  3. $\displaystyle\,p\,=\,1,\,\frac{-2}{7}$
  4. $\displaystyle\,p\,=\,1,\,\frac{2}{7}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

7p^2 - 5p - 2 = 0. Using the quadratic formula or factoring: 7p^2 - 7p + 2p - 2 = 0 -> 7p(p-1) + 2(p-1) = 0 -> (7p+2)(p-1) = 0. Roots are p = 1 and p = -2/7.

Multiple choice

Roots of the quadratic equation $4x^2\,+\,7x\,+\,2\,=\,0$, are
  1. $\displaystyle\,x\,=\,\frac{-7\,\pm\,\sqrt{11}}{8}$
  2. $\displaystyle\,x\,=\,\frac{-7\,\pm\,\sqrt{-17}}{8}$
  3. $\displaystyle\,x\,=\,\frac{-7\,\pm\,\sqrt{17}}{8}$
  4. $\displaystyle\,x\,=\,\frac{-4\,\pm\,\sqrt{17}}{8}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the quadratic formula x = (-7 +/- sqrt(49 - 4*4*2)) / 8 = (-7 +/- sqrt(49 - 32)) / 8 = (-7 +/- sqrt(17)) / 8.

Multiple choice

Find the roots of following quadratic equation $9s^2\,-\,4\,=\,-6s $
  1. $\displaystyle\,s\,=\,\frac{-1\,\pm\,\sqrt5}{3}$
  2. $\displaystyle\,s\,=\,\frac{-1\,\pm\,\sqrt5}{2}$
  3. $\displaystyle\,s\,=\,\frac{-5\,\pm\,\sqrt5}{3}$
  4. $\displaystyle\,s\,=\,\frac{-1\,\pm\,\sqrt5}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

9s^2 + 6s - 4 = 0. Using quadratic formula: s = (-b +/- sqrt(b^2 - 4ac)) / 2a. s = (-6 +/- sqrt(36 - 4*9*(-4))) / 18 = (-6 +/- sqrt(36 + 144)) / 18 = (-6 +/- sqrt(180)) / 18 = (-6 +/- 6*sqrt(5)) / 18 = (-1 +/- sqrt(5)) / 3.

Multiple choice

Roots of quadratic equation $3y^2\,+\,7y\,+\,4\,=\,0$ are
  1. $\displaystyle\,y\,=\,-1,\,\frac{-4}{3}$
  2. $\displaystyle\,y\,=\,2,\,\frac{-4}{3}$
  3. $\displaystyle\,y\,=\,-2,\,\frac{-4}{3}$
  4. $\displaystyle\,y\,=\,-1,\,\frac{-4}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Factoring 3y^2 + 7y + 4 = 0 gives (3y + 4)(y + 1) = 0. Setting each factor to zero yields y = -4/3 and y = -1.

Multiple choice

Roots of quadratic equation $2n^2\,+\,5n\,+\,2\,=\,0$ are
  1. $\displaystyle\,n\,=\,-8,\,\frac{-1}{2}$
  2. $\displaystyle\,n\,=\,-2,\,\frac{-1}{5}$
  3. $\displaystyle\,n\,=\,-2,\,\frac{1}{2}$
  4. $\displaystyle\,n\,=\,-2,\,\frac{-1}{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Factor the quadratic equation 2n^2 + 5n + 2 = 0. (2n + 1)(n + 2) = 0. The roots are n = -1/2 and n = -2.