To find the values of \(x\) that make the following equations equal to \((x + 4)(x - 2)\), we first need to expand the right side and set up an equation.
The expression \((x + 4)(x - 2)\) can be expanded as follows:
\[ (x + 4)(x - 2) = x^2 - 2x + 4x - 8 = x^2 + 2x - 8 \]
Next, we will set this equal to zero:
\[ x^2 + 2x - 8 = 0 \]
Now, we solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 2\), and \(c = -8\).
First, we calculate the discriminant:
\[ b^2 - 4ac = 2^2 - 4(1)(-8) = 4 + 32 = 36 \]
Now we can find the roots:
\[ x = \frac{-2 \pm \sqrt{36}}{2 \cdot 1} = \frac{-2 \pm 6}{2} \]
Calculating the two possible values for \(x\):
-
\(x = \frac{-2 + 6}{2} = \frac{4}{2} = 2\)
-
\(x = \frac{-2 - 6}{2} = \frac{-8}{2} = -4\)
Thus, the solutions are:
\[ x = 2 \quad \text{and} \quad x = -4 \]
Looking at the options provided:
- x = -2 and x = 4
- x = -2 and x = -4
- x = 2 and x = 4
- x = -4 and x = 2
The correct values are:
\[ \boxed{-4 \text{ and } 2} \]
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