Multiple choice

$\alpha$ and $\beta$ are two numbers such that $\alpha + \beta = 6$, $\alpha - \beta = 8$. Then, $\alpha$ and $\beta$ are the roots of the quadratic equation __________.

  1. $x^2 + 6x + 7 = 0$
  2. $x^2 - 6x - 7 = 0$
  3. $x^2 + 6x - 8 = 0$
  4. $x^2- 6x + 8 = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A quadratic equation with roots alpha and beta is given by x^2 - (alpha + beta)x + alpha * beta = 0. We are given the sum of the roots alpha + beta = 6. Solving the system alpha + beta = 6 and alpha - beta = 8 gives alpha = 7 and beta = -1, which means the product of the roots is -7. Substituting these values into the formula yields x^2 - 6x - 7 = 0.

AI explanation

Adding the equations alpha + beta = 6 and alpha - beta = 8 gives 2*alpha = 14, so alpha = 7. Substituting alpha into the first equation yields 7 + beta = 6, meaning beta = -1. A quadratic equation with these roots is built as x^2 - (sum)x + (product) = 0, resulting in x^2 - 6x - 7 = 0.

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