To solve the equation x² – 30x + 12 = 0 using the Zero Product Property, we need to factor the quadratic expression on the left-hand side. The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero.
First, let’s factor the quadratic equation. We need to find two numbers that multiply to the constant term (+12) and add up to the coefficient of the linear term (-30). After checking the possible pairs, we can see that it doesn’t factor nicely. Instead, we can apply the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, a = 1, b = -30, and c = 12. Plugging these values into the formula, we get:
x = (30 ± √((-30)² – 4 * 1 * 12)) / (2 * 1)
This simplifies to:
x = (30 ± √(900 – 48)) / 2
x = (30 ± √852) / 2
Now, we can simplify √852:
√852 = √(4 * 213) = 2√213
So, substituting back in, we have:
x = (30 ± 2√213) / 2
Breaking that down gives:
x = 15 ± √213
Thus, the solutions to the equation x² – 30x + 12 = 0 are:
x = 15 + √213 and x = 15 – √213.