Question

solve using factoring.
$x^2 - 4x - 17 = 4$
select the correct response:
13, -3
-7, 3
-7
-3, 7

Explanation:

Step1: Rearrange the equation

First, we need to set the equation to standard quadratic form \(ax^2 + bx + c = 0\). Subtract 4 from both sides of the equation \(x^2 - 4x - 17 = 4\):
\(x^2 - 4x - 17 - 4 = 0\)
\(x^2 - 4x - 21 = 0\)

Step2: Factor the quadratic

We need to find two numbers that multiply to \(-21\) (the product of \(a = 1\) and \(c = -21\)) and add up to \(-4\) (the coefficient \(b\)). The numbers are \(-7\) and \(3\) because \(-7\times3=-21\) and \(-7 + 3=-4\). So we can factor the quadratic as:
\((x - 7)(x + 3)=0\) Wait, no, wait. Wait, let's check again. Wait, if we have \(x^2 - 4x - 21\), the factors should be \((x - 7)(x + 3)\)? Wait, no, \((x - 7)(x + 3)=x^2+3x - 7x - 21=x^2 - 4x - 21\), yes. Wait, no, wait, when we set each factor equal to zero:
\(x - 7 = 0\) gives \(x = 7\)
\(x + 3 = 0\) gives \(x = -3\)
Wait, that's not one of the options. Wait, maybe I made a mistake. Wait, the options are 13, -3; -7, 3; -7; -3,7. Wait, let's re - factor. Wait, maybe the factors are \((x + 3)(x - 7)\) no, wait, maybe I messed up the signs. Wait, let's use the quadratic formula to check. The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) for \(ax^2+bx + c = 0\). Here, \(a = 1\), \(b=-4\), \(c=-21\). Then \(b^2 - 4ac=(-4)^2-4\times1\times(-21)=16 + 84 = 100\). Then \(\sqrt{100}=10\). So \(x=\frac{4\pm10}{2}\). So \(x=\frac{4 + 10}{2}=\frac{14}{2}=7\) and \(x=\frac{4-10}{2}=\frac{-6}{2}=-3\). Wait, but 7 and -3 are not in the options. Wait, the options have -3,7. Oh! The option is -3,7. So the correct factoring should lead to \(x = 7\) and \(x=-3\), which is the option -3,7.

Wait, let's re - do the factoring. \(x^2-4x - 21\). We need two numbers that multiply to -21 and add to -4. The numbers are -7 and 3? No, -7+3=-4, but when we factor, it's \((x - 7)(x + 3)\), but when we solve, \(x - 7 = 0\) gives \(x = 7\), \(x+3 = 0\) gives \(x=-3\). So the solutions are \(x = 7\) and \(x=-3\), which is the option -3,7.

Answer:

-3, 7 (the option with -3,7)