Solve for x y=4x^2-2x-5

$y = 4 x^{2} - 2 x - 5$

Step 1

Rewrite the equation as $4 x^{2} - 2 x - 5 = y$ .

$4 x^{2} - 2 x - 5 = y$

Step 2

Subtract $y$ from both sides of the equation.

$4 x^{2} - 2 x - 5 - y = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = - 2$ , and $c = - 5 - y$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot (- 5 - y)}}{2 \cdot 4}$

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16 \cdot (- 5 - y)}}{2 \cdot 4}$

Apply the distributive property.

$x = \frac{2 \pm \sqrt{4 - 16 \cdot - 5 - 16 (- y)}}{2 \cdot 4}$

Multiply $- 16$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 80 - 16 (- y)}}{2 \cdot 4}$

Multiply $- 1$ by $- 16$ .

$x = \frac{2 \pm \sqrt{4 + 80 + 16 y}}{2 \cdot 4}$

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84 + 16 y}}{2 \cdot 4}$

Factor $4$ out of $84 + 16 y$ .

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21) + 16 y}}{2 \cdot 4}$

Factor $4$ out of $16 y$ .

$x = \frac{2 \pm \sqrt{4 (21) + 4 (4 y)}}{2 \cdot 4}$

Factor $4$ out of $4 (21) + 4 (4 y)$ .

$x = \frac{2 \pm \sqrt{4 (21 + 4 y)}}{2 \cdot 4}$

Rewrite $4 (21 + 4 y)$ as $2^{2} (21 + 2^{2} y)$ .

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} (21 + 4 y)}}{2 \cdot 4}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} (21 + 2^{2} y)}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21 + 2^{2} y}}{2 \cdot 4}$

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

$x = \frac{2 \pm 2 \sqrt{21 + 4 y}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{2 \pm 2 \sqrt{21 + 4 y}}{8}$

Simplify $\frac{2 \pm 2 \sqrt{21 + 4 y}}{8}$ .

$x = \frac{1 \pm \sqrt{21 + 4 y}}{4}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{21 + 4 y}}{4}$

$x = \frac{1 - \sqrt{21 + 4 y}}{4}$

Solve for x y=4x^2-2x-5

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