Multiple choice

In the following, determine whether the given quadratic equations have real roots and if so, find the roots : $16x^2$ = 24x + 1

  1. $\text{not real}$
  2. $real,\dfrac{3 \, \pm \, \sqrt{10}}{4}$
  3. $\dfrac{3 \, \pm \, \sqrt{10}}{2}$
  4. $\dfrac{3 \, \pm \, \sqrt{13}}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

16x^2 - 24x - 1 = 0. Discriminant D = b^2 - 4ac = (-24)^2 - 4(16)(-1) = 576 + 64 = 640. Roots = (-b +/- sqrt(D)) / 2a = (24 +/- sqrt(640)) / 32 = (24 +/- 8*sqrt(10)) / 32 = (3 +/- sqrt(10)) / 4.

AI explanation

Rewrite the equation as 16x^2 - 24x - 1 = 0, giving a = 16, b = -24, and c = -1. Using the quadratic formula, x = (-b ± sqrt(b^2 - 4ac)) / (2a), we calculate the discriminant as (-24)^2 - 4(16)(-1) = 576 + 64 = 640. The square root of 640 simplifies to 8(sqrt(10)), so the roots are (24 ± 8(sqrt(10))) / 32. Dividing the numerator and denominator by 8 gives the real roots as (3 ± sqrt(10)) / 4.

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