Problem 3 - College Algebra

Question

Solve each equation in Exercises \(1-14\) by factoring. $$x^{2}=8 x-15$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3, 5\).

Step 1: Write the equation in standard form

First, move all terms to one side of the equation to put it in standard form. Subtract \(8x\) from both sides and add \(15\) to both sides gives \(x^2 - 8x + 15 =0\)

Step 2: Factoring

To factorize it, find two numbers that multiply to \(ac = 15\) and adds up to \(b = -8\). The numbers -3 and -5 satisfy both those conditions. Therefore, break down the -8x into -3x and -5x: \(x^2 - 3x - 5x + 15 =0\)

Step 3: Factor by Grouping

Group the terms and factor by grouping: \(x(x - 3) - 5(x - 3) = 0\). Now, notice that \((x - 3)\) is a common factor. Factoring out gives \((x - 3)(x - 5) = 0\)

Step 4: Solve for x

Set each factor to zero and solve for \(x\): From \(x - 3 = 0\), \(x = 3\), From \(x - 5 = 0\), \(x = 5\). So, the solution to the equation is \(x = 3, 5\)