Solve Using the Quadratic Formula 5x-4x^2=1

$5 x - 4 x^{2} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$5 x - 4 x^{2} - 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 4$ , $b = 5$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 4 \cdot - 1)}}{2 \cdot - 4}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 4 \cdot - 1}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot - 1$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{- 5 \pm \sqrt{25 + 16 \cdot - 1}}{2 \cdot - 4}$

Multiply $16$ by $- 1$ .

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

Subtract $16$ from $25$ .

$x = \frac{- 5 \pm \sqrt{9}}{2 \cdot - 4}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 5 \pm \sqrt{3^{2}}}{2 \cdot - 4}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 5 \pm 3}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{- 5 \pm 3}{- 8}$

Simplify $\frac{- 5 \pm 3}{- 8}$ .

$x = \frac{5 \pm 3}{8}$

Step 5

The final answer is the combination of both solutions.

$x = 1, \frac{1}{4}$