Question

1. using the information and diagram provided, solve for x. (hint: get equal to zero and then factor!)

Explanation:

Step1: Use the property of alternate exterior angles

Since \(\angle1\) and \(\angle2\) are alternate exterior angles, \(m\angle1 = m\angle2\). So we have the equation \(x^{2}-19=-7x + 41\).

Step2: Rearrange the equation to standard quadratic form

Move all terms to one side: \(x^{2}+7x-19 - 41=0\), which simplifies to \(x^{2}+7x - 60=0\).

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \(-60\) and add up to \(7\). The numbers are \(12\) and \(-5\). So the factored form is \((x + 12)(x-5)=0\).

Step4: Solve for \(x\)

Set each factor equal to zero:

• \(x+12 = 0\), then \(x=-12\).
• \(x - 5=0\), then \(x = 5\).

Answer:

\(x=-12\) or \(x = 5\)