Question

solve for all values of x by factoring. x² + 5x - 81 = 5x

Explanation:

Step1: Simplify the equation

Subtract \(5x\) from both sides of the equation \(x^{2}+5x - 81=5x\) to get a quadratic equation in standard form.
\(x^{2}+5x - 5x-81 = 5x - 5x\)
Simplifying both sides, we have \(x^{2}-81 = 0\)

Step2: Factor the quadratic expression

Notice that \(x^{2}-81\) is a difference of squares. The formula for factoring a difference of squares is \(a^{2}-b^{2}=(a + b)(a - b)\). Here, \(a=x\) and \(b = 9\) (since \(81=9^{2}\)). So we can factor \(x^{2}-81\) as \((x + 9)(x - 9)\).
So the equation becomes \((x + 9)(x - 9)=0\)

Step3: Solve for x using the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
For \((x + 9)(x - 9)=0\), we set each factor equal to zero:

• If \(x+9 = 0\), then \(x=-9\) (by subtracting 9 from both sides of the equation \(x + 9=0\))
• If \(x - 9=0\), then \(x = 9\) (by adding 9 to both sides of the equation \(x - 9=0\))

Answer:

\(x=-9\) or \(x = 9\)