Question Find values of $x$ such that $(x+7)(x-5)=13$ . The solution set is $x=2, x=-12$ .

Studdy Solution

STEP 1

Assumptions1. The given equations are $y_{1}=(x+7)$ , $y_{}=(x-5)$ , and $y_{1} y_{}=13$ . . We are asked to find all values of $x$ that satisfy these conditions.

STEP 2

From the given conditions, we know that $y_{1} y_{2}=13$ . We can substitute $y_{1}=(x+7)$ and $y_{2}=(x-5)$ into this equation.
$y_{1} y_{2} = (x+7)(x-5)$

STEP 3

Now, we set the equation equal to13.
$(x+7)(x-5) =13$

STEP 4

Next, we expand the left side of the equation.
$x^2 +2x -35 =13$

STEP 5

Subtract13 from both sides of the equation to set it equal to zero.
$x^2 +2x -48 =0$

STEP 6

Now, we factor the quadratic equation.
$(x -6)(x +8) =0$

STEP 7

Setting each factor equal to zero gives the solutions for $x$ .
$x -6 =0 \quad \text{or} \quad x + =0$

STEP 8

olving each equation gives the solutions for $x$ .
$x =6 \quad \text{or} \quad x = -8$ The solution set is $\{6, -8\}$ .

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