Solving Quadratic Equations with the Quadratic Formula
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 \( d = \frac{-1}{2} \).
