Write as a Function of k 4x^2+5x+k=0

$4 x^{2} + 5 x + k = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 4$ , $b = 5$ , and $c = k$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (4 \cdot k)}}{2 \cdot 4}$

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot k}}{2 \cdot 4}$

Multiply $- 4$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 k}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 k}}{8}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot k}}{2 \cdot 4}$

Multiply $- 4$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 k}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 k}}{8}$

Change the $\pm$ to $+$ .

$x = \frac{- 5 + \sqrt{25 - 16 k}}{8}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + \sqrt{25 - 16 k}}{8}$

Factor $- 1$ out of $\sqrt{25 - 16 k}$ .

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{25 - 16 k})}{8}$

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{25 - 16 k})$ .

$x = \frac{- 1 (5 - \sqrt{25 - 16 k})}{8}$

Move the negative in front of the fraction.

$x = - \frac{5 - \sqrt{25 - 16 k}}{8}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot k}}{2 \cdot 4}$

Multiply $- 4$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 k}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 k}}{8}$

Change the $\pm$ to $-$ .

$x = \frac{- 5 - \sqrt{25 - 16 k}}{8}$

Factor $- 1$ out of $- 5 - \sqrt{25 - 16 k}$ .

Reorder $25$ and $- 16 k$ .

$x = \frac{- 5 - \sqrt{- 16 k + 25}}{8}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - \sqrt{- 16 k + 25}}{8}$

Factor $- 1$ out of $- \sqrt{- 16 k + 25}$ .

$x = \frac{- 1 \cdot 5 - (\sqrt{- 16 k + 25})}{8}$

Factor $- 1$ out of $- 1 (5) - (\sqrt{- 16 k + 25})$ .

$x = \frac{- 1 (5 + \sqrt{- 16 k + 25})}{8}$

Rewrite $- 1 (5 + \sqrt{- 16 k + 25})$ as $- (5 + \sqrt{- 16 k + 25})$ .

$x = \frac{- (5 + \sqrt{- 16 k + 25})}{8}$

Move the negative in front of the fraction.

$x = - \frac{5 + \sqrt{- 16 k + 25}}{8}$

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{25 - 16 k}}{8}$

$x = - \frac{5 + \sqrt{- 16 k + 25}}{8}$

Write as a Function of k 4x^2+5x+k=0

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