Finding Solutions Using the Zero Product Property
To solve the equation x² – 30x + 12 = 0 using the zero product property, we first need to break it down into a suitable format. The zero product property states that if the product of two factors equals zero, then at least one of those factors must be zero.
Step 1: Factor the Quadratic Equation
We will start by factoring the quadratic expression x² – 30x + 12. To do this, we need to find two numbers that multiply to 12 (the constant term) and add up to -30 (the coefficient of the x term).
After analyzing the factors of 12, we see that the numbers 1 and 12 are too small and so are all other combinations of factors that could sum to -30, meaning the quadratic does not factor nicely. Thus, we can turn to the quadratic formula.
Step 2: Use the Quadratic Formula
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)For our equation x² – 30x + 12 = 0, we have:
- a = 1
- b = -30
- c = 12
Now plug the values into the formula:
x = (30 ± √((-30)² - 4 * 1 * 12)) / (2 * 1)Calculating the discriminant:
(-30)² - 4 * 1 * 12 = 900 - 48 = 852Continuing with the formula:
x = (30 ± √852) / 2Step 3: Simplifying the Result
The square root of 852 simplifies to:
√852 = 2√213So, we have:
x = (30 ± 2√213) / 2Which simplifies to:
x = 15 ± √213