Solve 2x^2-9=x

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

Solve for x

$$x = \frac{\sqrt{73} + 1}{4} \approx 2.386000936$$

$$x=\frac{1-\sqrt{73}}{4}\approx -1.886000936$$

Steps Using the Quadratic Formula

Subtract x from both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-9\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-9\right)}}{2\times 2}$$

Multiply -8 times -9.

$$x=\frac{-\left(-1\right)±\sqrt{1+72}}{2\times 2}$$

Add 1 to 72.

$$x=\frac{-\left(-1\right)±\sqrt{73}}{2\times 2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{73}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{1±\sqrt{73}}{4}$$

Now solve the equation x=\frac{1±\sqrt{73}}{4} when ± is plus. Add 1 to \sqrt{73}.

$$x=\frac{\sqrt{73}+1}{4}$$

Now solve the equation x=\frac{1±\sqrt{73}}{4} when ± is minus. Subtract \sqrt{73} from 1.

$$x=\frac{1-\sqrt{73}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{73}+1}{4} x=\frac{1-\sqrt{73}}{4}$$

Steps for Completing the Square

Subtract x from both sides.

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

Add 9 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{9}{2}+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{9}{2}+\frac{1}{16}$$

Add \frac{9}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{73}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{73}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{73}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{73}}{4} x-\frac{1}{4}=-\frac{\sqrt{73}}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{\sqrt{73}+1}{4} x=\frac{1-\sqrt{73}}{4}$$