Solve 1.5=x^2-x

$$1.5= { x }^{ 2 } -x$$

Solve for x

$$x = \frac{\sqrt{7} + 1}{2} \approx 1.822875656$$

$$x=\frac{1-\sqrt{7}}{2}\approx -0.822875656$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=1.5$$

Subtract 1.5 from both sides.

$$x^{2}-x-1.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -1.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1.5\right)}}{2}$$

Multiply -4 times -1.5.

$$x=\frac{-\left(-1\right)±\sqrt{1+6}}{2}$$

Add 1 to 6.

$$x=\frac{-\left(-1\right)±\sqrt{7}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{7}}{2}$$

Now solve the equation x=\frac{1±\sqrt{7}}{2} when ± is plus. Add 1 to \sqrt{7}.

$$x=\frac{\sqrt{7}+1}{2}$$

Now solve the equation x=\frac{1±\sqrt{7}}{2} when ± is minus. Subtract \sqrt{7} from 1.

$$x=\frac{1-\sqrt{7}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{7}+1}{2} x=\frac{1-\sqrt{7}}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=1.5$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=1.5+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=1.5+\frac{1}{4}$$

Add 1.5 to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{7}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{7}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{7}}{2} x-\frac{1}{2}=-\frac{\sqrt{7}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{7}+1}{2} x=\frac{1-\sqrt{7}}{2}$$