Precalculus Examples

$4 x^{2} = 10 x - 9$

Step 1

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $10 x$ from both sides of the equation.

$4 x^{2} - 10 x = - 9$

Step 1.2

Add $9$ to both sides of the equation.

$4 x^{2} - 10 x + 9 = 0$

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 $a = 4$ , $b = - 10$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (4 \cdot 9)}}{2 \cdot 4}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{10 \pm \sqrt{100 - 16 \cdot 9}}{2 \cdot 4}$

Step 4.1.2.2

Multiply $- 16$ by $9$ .

$x = \frac{10 \pm \sqrt{100 - 144}}{2 \cdot 4}$

Step 4.1.3

Subtract $144$ from $100$ .

$x = \frac{10 \pm \sqrt{- 44}}{2 \cdot 4}$

Step 4.1.4

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 44}}{2 \cdot 4}$

Step 4.1.5

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot 4}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{44}}{2 \cdot 4}$

Step 4.1.7

Rewrite $44$ as $2^{2} \cdot 11$ .

Tap for more steps...

Step 4.1.7.1

Factor $4$ out of $44$ .

$x = \frac{10 \pm i \cdot \sqrt{4 (11)}}{2 \cdot 4}$

Step 4.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot 4}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (2 \sqrt{11})}{2 \cdot 4}$

Step 4.1.9

Move $2$ to the left of $i$ .

$x = \frac{10 \pm 2 i \sqrt{11}}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$x = \frac{10 \pm 2 i \sqrt{11}}{8}$

Step 4.3

Simplify $\frac{10 \pm 2 i \sqrt{11}}{8}$ .

$x = \frac{5 \pm i \sqrt{11}}{4}$