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

Solve the following quadratic equation by factorization method: $\sqrt{5}x^2+2x-3\sqrt{5}=0$
  1. $\left \{ -\sqrt{5}, \dfrac{3}{\sqrt{5}} \right \}$
  2. $\left \{ \sqrt{5}, \dfrac{3}{\sqrt{5}} \right \}$
  3. $\left \{ -\sqrt{5}, \dfrac{-3}{\sqrt{5}} \right \}$

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

The equation is sqrt(5)x^2 + 2x - 3sqrt(5) = 0. Using the quadratic formula or factoring: sqrt(5)x^2 + 5x - 3x - 3sqrt(5) = 0, which is sqrt(5)x(x + sqrt(5)) - 3(x + sqrt(5)) = 0. This gives (sqrt(5)x - 3)(x + sqrt(5)) = 0, resulting in x = -sqrt(5) and x = 3/sqrt(5).

Multiple choice

Solve the following quadratic equations by factorization method: $3(x^2-6)=x(x+7)-3$
  1. $\left \{ -\dfrac{3}{2}, 5 \right \}$ 
  2. $\left \{ \dfrac{3}{2}, 5 \right \}$ 
  3. $\left \{ -\dfrac{3}{2}, -5 \right \}$ 

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

Expanding both sides of the equation gives 3x^2 - 18 = x^2 + 7x - 3. Rearranging all terms to one side yields the quadratic equation 2x^2 - 7x - 15 = 0. Factoring this expression gives (2x + 3)(x - 5) = 0, which has the roots x = -3/2 and x = 5.

Multiple choice

Solve the following quadratic equations by factorization method: $\dfrac{x}{x+1}+\dfrac{x+1}{x}=\dfrac{34}{15}$
  1. $\left \{ -\dfrac{5}{2}, \dfrac{3}{2} \right \}$ 
  2. $\left \{ \dfrac{5}{2}, \dfrac{3}{2} \right \}$ 
  3. $\left \{ -\dfrac{5}{2}, \dfrac{-3}{2} \right \}$ 

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

Let y = x/(x+1). The equation becomes y + 1/y = 34/15. Solving y^2 - (34/15)y + 1 = 0 gives y = 5/3 or y = 3/5. Solving x/(x+1) = 5/3 gives x = -5/2, and x/(x+1) = 3/5 gives x = 3/2.

Multiple choice

Solve the following quadratic equations by factorization method: $2(x+1)^2-5(x+1)=12$
  1. $\left \{ -\dfrac{5}{2}, 3 \right \}$ 
  2. $\left \{ \dfrac{5}{2}, 3 \right \}$ 
  3. $\left \{ -\dfrac{5}{2}, -3 \right \}$ 

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

Let u = x+1. The equation becomes 2u^2 - 5u - 12 = 0. Factoring: 2u^2 - 8u + 3u - 12 = 0, 2u(u-4) + 3(u-4) = 0, (2u+3)(u-4) = 0. So u = -3/2 or u = 4. Since u = x+1, x = -3/2 - 1 = -5/2 or x = 4 - 1 = 3.

Multiple choice

Solve the following equations: $\dfrac {x + \sqrt {x^{2} - 1}}{x - \sqrt {x^{2} - 1}} + \dfrac {x - \sqrt {x^{2} - 1}}{x + \sqrt {x^{2} - 1}} = 98$
  1. $\pm 1$
  2. $\pm5$
  3. $\pm 10$
  4. $\pm 4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let y = (x + sqrt(x^2-1)) / (x - sqrt(x^2-1)). Then the equation is y + 1/y = 98. Solving y^2 - 98y + 1 = 0 gives y = 49 +/- sqrt(2400) = 49 +/- 20*sqrt(6). Alternatively, note that (x + sqrt(x^2-1))^2 = x^2 + x^2 - 1 + 2x*sqrt(x^2-1) = 2x^2 - 1 + 2x*sqrt(x^2-1). Testing x=5: (5+sqrt(24))/(5-sqrt(24)) = (5+sqrt(24))^2 = 25+24+10*sqrt(24) = 49+20*sqrt(6). The sum is 98.

Multiple choice

Solve the following equations: $\sqrt {x^{2} + ax - 1} - \sqrt {x^{2} + bx - 1} = \sqrt {a} - \sqrt {b}$.
  1. $-2, \dfrac {(\sqrt {a} - \sqrt {b})^{2} + 3}{(\sqrt {a} + \sqrt {b})^{2} - 3}$
  2. $-1, \dfrac {(\sqrt {a} - \sqrt {b})^{2} - 4}{(\sqrt {a} + \sqrt {b})^{2} + 4}$
  3. $\dfrac {1}{2}, \dfrac {(\sqrt {a} - \sqrt {b})^{2} + 4}{(\sqrt {a} + \sqrt {b})^{2} - 4}$
  4. $1, \dfrac {(\sqrt {a} - \sqrt {b})^{2} + 4}{(\sqrt {a} + \sqrt {b})^{2} - 4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let u = sqrt(x^2 + ax - 1) and v = sqrt(x^2 + bx - 1). The equation is u - v = sqrt(a) - sqrt(b). Squaring both sides and simplifying leads to the solution set including 1 and the expression in option D.