Question

Sophia was solving the following quadratic equation by factoring: `x^{2}-8x-20=0`. Which choice shows the correctly factored form of this equation?

(x - 5) (x - 4) = 0
(x - 5) (x + 4) = 0
(x - 10) (x - 2) = 0
(x - 10) (x + 2) = 0

Answer 1

To factor the quadratic equation \(x^{2}-8x-20=0\), we need to find two numbers that multiply to \(-20\) (the constant term) and add to \(-8\) (the coefficient of the \(x\) term).

The factors of \(-20\) include:
- \(1\) and \(-20\)
- \(-1\) and \(20\)
- \(2\) and \(-10\)
- \(-2\) and \(10\)
- \(4\) and \(-5\)
- \(-4\) and \(5\)

Among these pairs, the pair \(-10\) and \(2\) adds up to \(-8\) (since \(-10 + 2 = -8\)).

Therefore, we can factor the quadratic as follows:

\[
x^{2} - 8x - 20 = (x - 10)(x + 2)
\]

Setting this equal to zero gives us:

\[
(x - 10)(x + 2) = 0
\]

So the correct factored form of the equation \(x^{2} - 8x - 20 = 0\) is:

\((x - 10)(x + 2) = 0\)

Thus, the correct choice is:

\((x - 10)(x + 2) = 0\)