Use factoring and the zero-product property to solve the following problems.
1.9r^2-30r+21=-4
To solve the equation 9r^2 - 30r + 21 = -4 using factoring and the zero-product property, follow these steps:
Step 1: Move all terms to one side of the equation to set it equal to zero:
9r^2 - 30r + 21 + 4 = 0
Simplify:
9r^2 - 30r + 25 = 0
Step 2: Factor the quadratic expression on the left side of the equation. To do this, we need to find two numbers that multiply to 9 * 25 = 225, and also add up to -30 (the coefficient of the middle term).
The numbers that satisfy these conditions are -15 and -15, since (-15) * (-15) = 225 and -15 + (-15) = -30.
So, we can rewrite the quadratic expression as:
(3r - 5)(3r - 5) = 0
Step 3: Solve for r by setting each factor equal to zero using the zero-product property:
3r - 5 = 0
Step 4: Solve the equation for r:
3r = 5
r = 5/3
Therefore, the solution to the equation 9r^2 - 30r + 21 = -4 is r = 5/3.