Solve -1=x^2-2.5x

$$-1= { x }^{ 2 } -2.5x$$

Solve for x

$$x=2$$

$$x=0.5$$

Steps Using the Quadratic Formula

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

$$x^{2}-2.5x=-1$$

Add 1 to both sides.

$$x^{2}-2.5x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2.5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2.5\right)±\sqrt{\left(-2.5\right)^{2}-4}}{2}$$

Square -2.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-2.5\right)±\sqrt{6.25-4}}{2}$$

Add 6.25 to -4.

$$x=\frac{-\left(-2.5\right)±\sqrt{2.25}}{2}$$

Take the square root of 2.25.

$$x=\frac{-\left(-2.5\right)±\frac{3}{2}}{2}$$

The opposite of -2.5 is 2.5.

$$x=\frac{2.5±\frac{3}{2}}{2}$$

Now solve the equation x=\frac{2.5±\frac{3}{2}}{2} when ± is plus. Add 2.5 to \frac{3}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{2.5±\frac{3}{2}}{2} when ± is minus. Subtract \frac{3}{2} from 2.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=2 x=\frac{1}{2}$$

Steps for Completing the Square

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

$$x^{2}-2.5x=-1$$

Divide -2.5, the coefficient of the x term, by 2 to get -1.25. Then add the square of -1.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2.5x+\left(-1.25\right)^{2}=-1+\left(-1.25\right)^{2}$$

Square -1.25 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-2.5x+1.5625=-1+1.5625$$

Add -1 to 1.5625.

$$x^{2}-2.5x+1.5625=0.5625$$

Factor x^{2}-2.5x+1.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1.25\right)^{2}=0.5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1.25\right)^{2}}=\sqrt{0.5625}$$

Simplify.

$$x-1.25=\frac{3}{4} x-1.25=-\frac{3}{4}$$

Add 1.25 to both sides of the equation.

$$x=2 x=\frac{1}{2}$$