Question

question 10 of 23
solve by using the quadratic equation.
$x^2 + 6x - 16 = 0$
select the correct response:
$x = 2$
$x = -8$
$x = \frac{6 + \sqrt{28}}{2}$
$x = \frac{6 - \sqrt{28}}{2}$
$x = \frac{-6 + \sqrt{28}}{2}$
$x = \frac{-6 - \sqrt{28}}{2}$
$x = 16$
$x = -4$

Explanation:

Step1: Recall quadratic formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Here, \(a = 1\), \(b = 6\), \(c=-16\).

Step2: Calculate discriminant

Discriminant \(D=b^2 - 4ac=(6)^2-4\times1\times(-16)=36 + 64 = 100\)? Wait, no, wait, original equation is \(x^2+6x - 16=0\), so \(a = 1\), \(b = 6\), \(c=-16\). Then \(b^2-4ac=36-4\times1\times(-16)=36 + 64 = 100\)? Wait, but in the options, it's \(\sqrt{28}\). Wait, maybe I made a mistake. Wait, no, let's check again. Wait, the equation is \(x^2+6x - 16=0\)? Wait, maybe the user made a typo? Wait, no, let's proceed with the quadratic formula. Wait, \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}=\frac{-6\pm\sqrt{36+64}}{2}=\frac{-6\pm\sqrt{100}}{2}=\frac{-6\pm10}{2}\). So when we take the plus sign: \(\frac{-6 + 10}{2}=\frac{4}{2}=2\). When we take the minus sign: \(\frac{-6-10}{2}=\frac{-16}{2}=-8\). Also, let's check the other options. Wait, the options have \(\sqrt{28}\), which is wrong. Wait, maybe the equation was supposed to be \(x^2+6x - 2=0\)? But according to the given equation \(x^2+6x - 16=0\), the solutions are \(x = 2\) and \(x=-8\). Also, let's check the last option: \(x = 16\) and \(x=-4\) don't satisfy. Let's check \(x = 2\): \(2^2+6\times2-16=4 + 12-16=0\). \(x=-8\): \((-8)^2+6\times(-8)-16=64-48-16=0\). So the correct solutions are \(x = 2\) and \(x=-8\).

Answer:

\(x = 2\), \(x = -8\)