Quantitative Aptitude

Algebra

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Algebra Questions

Multiple choice

Which of the following equations has two equal real roots?
  1. $2x^2-3\sqrt{2}x+\cfrac{9}{4}=0$
  2. $x^2 + x + 6 = 0$
  3. $x^2 + 3x + 2 \sqrt{2} = 0$
  4. $5x^2 + 3x + 1 = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a quadratic equation to have equal real roots, the discriminant D = b^2 - 4ac must be zero. For 2x^2 - 3sqrt(2)x + 9/4 = 0, D = (-3sqrt(2))^2 - 4(2)(9/4) = 18 - 18 = 0.

Multiple choice

If the coefficient of $x^2$ and the constant term have the same sign and if the coefficient of $x$ term is zero, then the quadratic equation has :

  1. no real roots

  2. one real root

  3. one real and one imaginary roots

  4. two real roots
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If the x term is zero, the equation is ax^2 + c = 0. If a and c have the same sign, then ax^2 = -c, so x^2 = -c/a. Since c and a have the same sign, -c/a is negative. The square of a real number cannot be negative, so there are no real roots.

Multiple choice

The equation $x^2 + 5 \sqrt{5}x-70 = 0$ has

  1. real and unequal roots

  2. real and equal roots

  3. both imaginary roots

  4. one real and one imaginary roots
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Discriminant D = b^2 - 4ac = (5*sqrt(5))^2 - 4(1)(-70) = 125 + 280 = 405. Since D > 0, the roots are real and unequal.