Solve 9(x)=2x^2+2x-1/2

$$9 ( x ) = \frac { 2 x ^ { 2 } + 2 x - 1 } { 2 }$$

Solve for x

$$x=\frac{\sqrt{66}}{2}+4\approx 8.062019202$$

$$x=-\frac{\sqrt{66}}{2}+4\approx -0.062019202$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$18x=2x^{2}+2x-1$$

Subtract 2x^{2} from both sides.

$$18x-2x^{2}=2x-1$$

Subtract 2x from both sides.

$$18x-2x^{2}-2x=-1$$

Combine 18x and -2x to get 16x.

$$16x-2x^{2}=-1$$

Add 1 to both sides.

$$16x-2x^{2}+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2x^{2}+16x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 16 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-2\right)}}{2\left(-2\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-2\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-16±\sqrt{256+8}}{2\left(-2\right)}$$

Add 256 to 8.

$$x=\frac{-16±\sqrt{264}}{2\left(-2\right)}$$

Take the square root of 264.

$$x=\frac{-16±2\sqrt{66}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-16±2\sqrt{66}}{-4}$$

Now solve the equation x=\frac{-16±2\sqrt{66}}{-4} when ± is plus. Add -16 to 2\sqrt{66}.

$$x=\frac{2\sqrt{66}-16}{-4}$$

Divide -16+2\sqrt{66} by -4.

$$x=-\frac{\sqrt{66}}{2}+4$$

Now solve the equation x=\frac{-16±2\sqrt{66}}{-4} when ± is minus. Subtract 2\sqrt{66} from -16.

$$x=\frac{-2\sqrt{66}-16}{-4}$$

Divide -16-2\sqrt{66} by -4.

$$x=\frac{\sqrt{66}}{2}+4$$

The equation is now solved.

$$x=-\frac{\sqrt{66}}{2}+4 x=\frac{\sqrt{66}}{2}+4$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$18x=2x^{2}+2x-1$$

Subtract 2x^{2} from both sides.

$$18x-2x^{2}=2x-1$$

Subtract 2x from both sides.

$$18x-2x^{2}-2x=-1$$

Combine 18x and -2x to get 16x.

$$16x-2x^{2}=-1$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2x^{2}+16x=-1$$

Divide both sides by -2.

$$\frac{-2x^{2}+16x}{-2}=-\frac{1}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{16}{-2}x=-\frac{1}{-2}$$

Divide 16 by -2.

$$x^{2}-8x=-\frac{1}{-2}$$

Divide -1 by -2.

$$x^{2}-8x=\frac{1}{2}$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=\frac{1}{2}+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=\frac{1}{2}+16$$

Add \frac{1}{2} to 16.

$$x^{2}-8x+16=\frac{33}{2}$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=\frac{33}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{\frac{33}{2}}$$

Simplify.

$$x-4=\frac{\sqrt{66}}{2} x-4=-\frac{\sqrt{66}}{2}$$

Add 4 to both sides of the equation.

$$x=\frac{\sqrt{66}}{2}+4 x=-\frac{\sqrt{66}}{2}+4$$