Problem 11 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by factoring. $$x^{2}-7 x+12=0$$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = 4\) and \(x = 3\).

Step 1: Factoring the quadratic equation

We want to factor the equation \(x^2 - 7x + 12 = 0\). We need to find two numbers that multiply to \(+12\) (the constant term) and add to \(-7\) (the coefficient of x). These numbers are -4 and -3. So the factored form would be \((x-4)(x-3) = 0\).

Step 2: Setting each factor equal to zero

Now that we have factored the equation, we set each factor equal to zero and solve for x, i.e., \(x-4 = 0\) and \(x-3 = 0\).

Step 3: Solving for x

Solving these two equations, we get: \(x = 4\) from \(x-4 = 0\), and \(x = 3\) from \(x-3 = 0\). So the solutions to the original quadratic equation are \(x = 4\) and \(x = 3\).