Question

use the quadratic formula to solve. express your answer in simplest form.
$4d^2 - 5d - 5 = 3d$

Explanation:

Step1: Rearrange to standard quadratic form

Subtract $3d$ from both sides to get $ax^2+bx+c=0$.
$4d^2 - 5d - 5 - 3d = 0$
$4d^2 - 8d - 5 = 0$

Step2: Identify a, b, c values

From $4d^2 - 8d - 5 = 0$, we have:
$a=4$, $b=-8$, $c=-5$

Step3: Apply quadratic formula

Quadratic formula: $d=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute $a,b,c$:
$d=\frac{-(-8)\pm\sqrt{(-8)^2-4(4)(-5)}}{2(4)}$

Step4: Simplify discriminant

Calculate the value under the square root.
$\sqrt{64 + 80} = \sqrt{144} = 12$

Step5: Simplify the expression

Substitute discriminant result and simplify.
$d=\frac{8\pm12}{8}$

Step6: Solve for two roots

Split into addition and subtraction cases.
$d=\frac{8+12}{8}=\frac{20}{8}=\frac{5}{2}$, $d=\frac{8-12}{8}=\frac{-4}{8}=-\frac{1}{2}$

Answer:

$d=\frac{5}{2}$ or $d=-\frac{1}{2}$