Question - Solving Quadratic Equations with the Quadratic Formula
Solution:
To use the quadratic formula to solve the equation, we first need to put the equation in standard form, which is $$ ax^2 + bx + c = 0 $$.The equation given in the image is:\[ 4d^2 - 5d - 5 = 3d \]First, we will move all terms to one side to set the equation to zero:\[ 4d^2 - 5d - 5 - 3d = 0 \]\[ 4d^2 - 8d - 5 = 0 \]Now we have a quadratic equation in standard form, where $$ a = 4 $$, $$ b = -8 $$, and $$ c = -5 $$.The quadratic formula is:\[ d = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]Let's plug the values of $$ a $$, $$ b $$, and $$ c $$ into the quadratic formula:\[ d = \frac{{-(-8) \pm \sqrt{{(-8)^2 - 4 \cdot 4 \cdot (-5)}}}}{2 \cdot 4} \]\[ d = \frac{{8 \pm \sqrt{{64 + 80}}}}{8} \]\[ d = \frac{{8 \pm \sqrt{{144}}}}{8} \]\[ d = \frac{{8 \pm 12}}{8} \]Now we have two potential solutions:\[ d = \frac{{8 + 12}}{8} = \frac{20}{8} = \frac{5}{2} \]\[ d = \frac{{8 - 12}}{8} = \frac{-4}{8} = \frac{-1}{2} \]Hence, the solutions to the given quadratic equation in simplest form are $$ d = \frac{5}{2} $$ and $$ d = \frac{-1}{2} $$.
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