Question

solve for z by factoring. 9z^2 + 3z - 20 = 0

Explanation:

Step1: Factor the quadratic equation

For the quadratic equation \(9z^{2}+3z - 20=0\), we need to find two numbers that multiply to \(9\times(- 20)=-180\) and add up to \(3\). The numbers are \(15\) and \(-12\). So we can rewrite the middle - term: \(9z^{2}+15z-12z - 20 = 0\). Then factor by grouping: \(3z(3z + 5)-4(3z + 5)=0\), which gives \((3z + 5)(3z-4)=0\).

Step2: Set each factor equal to zero

If \(3z + 5=0\), then \(3z=-5\), and \(z=-\frac{5}{3}\). If \(3z - 4=0\), then \(3z=4\), and \(z=\frac{4}{3}\).

Answer:

\(z =-\frac{5}{3},z=\frac{4}{3}\)