Find All Complex Solutions x^2-5x=-9/4

$x^{2} - 5 x = - \frac{9}{4}$

Step 1

Add $\frac{9}{4}$ to both sides of the equation.

$x^{2} - 5 x + \frac{9}{4} = 0$

Step 2

Multiply through by the least common denominator $4$ , then simplify.

Apply the distributive property.

$4 x^{2} + 4 (- 5 x) + 4 (\frac{9}{4}) = 0$

Simplify.

Multiply $- 5$ by $4$ .

$4 x^{2} - 20 x + 4 (\frac{9}{4}) = 0$

Cancel the common factor of $4$ .

Cancel the common factor.

$4 x^{2} - 20 x + 4 (\frac{9}{4}) = 0$

Rewrite the expression.

$4 x^{2} - 20 x + 9 = 0$

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 $a = 4$ , $b = - 20$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (4 \cdot 9)}}{2 \cdot 4}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 9$ .

Multiply $- 4$ by $4$ .

$x = \frac{20 \pm \sqrt{400 - 16 \cdot 9}}{2 \cdot 4}$

Multiply $- 16$ by $9$ .

$x = \frac{20 \pm \sqrt{400 - 144}}{2 \cdot 4}$

Subtract $144$ from $400$ .

$x = \frac{20 \pm \sqrt{256}}{2 \cdot 4}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{20 \pm \sqrt{16^{2}}}{2 \cdot 4}$

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

$x = \frac{20 \pm 16}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{20 \pm 16}{8}$

Simplify $\frac{20 \pm 16}{8}$ .

$x = \frac{5 \pm 4}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{9}{2}, \frac{1}{2}$