Question

use the quadratic formula to solve. express your answer in simplest form. 4d² - 5d - 5 = 3d

Explanation:

Step1: Rearrange into standard form

First, we need to rewrite the equation \(4d^{2}-5d - 5=3d\) in the standard quadratic form \(ax^{2}+bx + c = 0\) (here we use \(d\) instead of \(x\)). Subtract \(3d\) from both sides:
\(4d^{2}-5d-3d - 5=0\)
Simplify the like terms: \(4d^{2}-8d - 5=0\)
Now we have \(a = 4\), \(b=- 8\), \(c=-5\)

Step2: Apply quadratic formula

The quadratic formula is \(d=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
Substitute \(a = 4\), \(b=-8\), \(c = - 5\) into the formula:
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-8)^{2}-4\times4\times(-5)\)
\(\Delta = 64 + 80=144\)
Then, \(d=\frac{-(-8)\pm\sqrt{144}}{2\times4}=\frac{8\pm12}{8}\)

Step3: Find two solutions

We have two cases:
Case 1: \(d=\frac{8 + 12}{8}=\frac{20}{8}=\frac{5}{2}\)
Case 2: \(d=\frac{8-12}{8}=\frac{-4}{8}=-\frac{1}{2}\)

Answer:

\(d=\frac{5}{2}\) or \(d =-\frac{1}{2}\)