Solve Using the Quadratic Formula 9x^2=10x+2

$9 x^{2} = 10 x + 2$

Step 1

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

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

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

Subtract $2$ from both sides of the equation.

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

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 $a = 9$ , $b = - 10$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 9 \cdot - 2$ .

Multiply $- 4$ by $9$ .

$x = \frac{10 \pm \sqrt{100 - 36 \cdot - 2}}{2 \cdot 9}$

Multiply $- 36$ by $- 2$ .

$x = \frac{10 \pm \sqrt{100 + 72}}{2 \cdot 9}$

Add $100$ and $72$ .

$x = \frac{10 \pm \sqrt{172}}{2 \cdot 9}$

Rewrite $172$ as $2^{2} \cdot 43$ .

Factor $4$ out of $172$ .

$x = \frac{10 \pm \sqrt{4 (43)}}{2 \cdot 9}$

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 43}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{43}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{10 \pm 2 \sqrt{43}}{18}$

Simplify $\frac{10 \pm 2 \sqrt{43}}{18}$ .

$x = \frac{5 \pm \sqrt{43}}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{43}}{9}$

Decimal Form:

$x = 1.28415983 \dots, - 0.17304872 \dots$