Solve each equation in Exercises 1 - 14 by factoring. $x^2=8x-15$

Verified step by step guidance

1. Rewrite the equation so that all terms are on one side, setting the equation equal to zero: \(x^2 - 8x + 15 = 0\).
2. Identify the quadratic expression to factor: \(x^2 - 8x + 15\).
3. Find two numbers that multiply to the constant term (15) and add up to the coefficient of the linear term (-8).
4. Use these two numbers to factor the quadratic into the form \((x - a)(x - b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
5. Set each factor equal to zero and solve for \(x\): \(x - a = 0\) and \(x - b = 0\).

Key Concepts

Rearranging Equations to Standard Form

To solve quadratic equations by factoring, first rewrite the equation so that one side equals zero. This involves moving all terms to one side, resulting in a standard form ax² + bx + c = 0, which is essential for applying factoring techniques.

Factoring Quadratic Expressions

Factoring involves expressing a quadratic polynomial as a product of two binomials. This process helps identify the roots of the equation by setting each factor equal to zero, simplifying the solution of the quadratic equation.

Zero Product Property

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to solve quadratic equations by setting each factor equal to zero and solving for the variable.