Solve Using the Quadratic Formula x(9x-5)=14

$x (9 x - 5) = 14$

Step 1

Move all terms to the left side of the equation and simplify.

Simplify the left side.

Simplify $x (9 x - 5)$ .

Simplify by multiplying through.

Apply the distributive property.

$x (9 x) + x \cdot - 5 = 14$

Reorder.

Rewrite using the commutative property of multiplication.

$9 x \cdot x + x \cdot - 5 = 14$

Move $- 5$ to the left of $x$ .

$9 x \cdot x - 5 \cdot x = 14$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$9 (x \cdot x) - 5 \cdot x = 14$

Multiply $x$ by $x$ .

$9 x^{2} - 5 \cdot x = 14$

$9 x^{2} - 5 x = 14$

Subtract $14$ from both sides of the equation.

$9 x^{2} - 5 x - 14 = 0$

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 $a = 9$ , $b = - 5$ , and $c = - 14$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (9 \cdot - 14)}}{2 \cdot 9}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot - 14}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot - 14$ .

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36 \cdot - 14}}{2 \cdot 9}$

Multiply $- 36$ by $- 14$ .

$x = \frac{5 \pm \sqrt{25 + 504}}{2 \cdot 9}$

Add $25$ and $504$ .

$x = \frac{5 \pm \sqrt{529}}{2 \cdot 9}$

Rewrite $529$ as $23^{2}$ .

$x = \frac{5 \pm \sqrt{23^{2}}}{2 \cdot 9}$

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

$x = \frac{5 \pm 23}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{5 \pm 23}{18}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{14}{9}, - 1$

Solve Using the Quadratic Formula x(9x-5)=14

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