Multiple choice

The number of quadratic equation $x^{2}+bx+c=0$ that have real and equal roots where coefficients $b$ and $c$ are from the set $\left{1,2,3,4,5\right}$ ($b$ and $c$ may be equal) is

  1. $4$
  2. $12$
  3. $2$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For real and equal roots, discriminant D = b^2 - 4c = 0, so b^2 = 4c. Given b, c in {1, 2, 3, 4, 5}: If b=1, c=1/4 (no). If b=2, c=1 (yes). If b=3, c=9/4 (no). If b=4, c=4 (yes). If b=5, c=25/4 (no). Only (2,1) and (4,4) work.

AI explanation

For the quadratic equation x^2 + bx + c = 0 to have real and equal roots, the discriminant must equal zero. Therefore, b^2 - 4c = 0, which means b^2 = 4c. Testing the available coefficients from the set {1, 2, 3, 4, 5}, if b = 2 then c = 1, and if b = 4 then c = 4, which are the only valid pairs. Therefore, there are exactly 2 such quadratic equations.

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