Solve -1/4x+15/8=-x^2-2x+3

$$- \frac{ 1 }{ 4 } x+ \frac{ 15 }{ 8 } = - { x }^{ 2 } -2x+3$$

Solve for x

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

$$x=\frac{1}{2}=0.5$$

Steps Using the Quadratic Formula

Add x^{2} to both sides.

$$-\frac{1}{4}x+\frac{15}{8}+x^{2}=-2x+3$$

Add 2x to both sides.

$$-\frac{1}{4}x+\frac{15}{8}+x^{2}+2x=3$$

Combine -\frac{1}{4}x and 2x to get \frac{7}{4}x.

$$\frac{7}{4}x+\frac{15}{8}+x^{2}=3$$

Subtract 3 from both sides.

$$\frac{7}{4}x+\frac{15}{8}+x^{2}-3=0$$

Subtract 3 from \frac{15}{8} to get -\frac{9}{8}.

$$\frac{7}{4}x-\frac{9}{8}+x^{2}=0$$

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.

$$x^{2}+\frac{7}{4}x-\frac{9}{8}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{7}{4} for b, and -\frac{9}{8} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{7}{4}±\sqrt{\left(\frac{7}{4}\right)^{2}-4\left(-\frac{9}{8}\right)}}{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{7}{4}±\sqrt{\frac{49}{16}-4\left(-\frac{9}{8}\right)}}{2}$$

Multiply -4 times -\frac{9}{8}.

$$x=\frac{-\frac{7}{4}±\sqrt{\frac{49}{16}+\frac{9}{2}}}{2}$$

Add \frac{49}{16} to \frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{7}{4}±\sqrt{\frac{121}{16}}}{2}$$

Take the square root of \frac{121}{16}.

$$x=\frac{-\frac{7}{4}±\frac{11}{4}}{2}$$

Now solve the equation x=\frac{-\frac{7}{4}±\frac{11}{4}}{2} when ± is plus. Add -\frac{7}{4} to \frac{11}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Now solve the equation x=\frac{-\frac{7}{4}±\frac{11}{4}}{2} when ± is minus. Subtract \frac{11}{4} from -\frac{7}{4} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -\frac{9}{2} by 2.

$$x=-\frac{9}{4}$$

The equation is now solved.

$$x=\frac{1}{2} x=-\frac{9}{4}$$

Steps for Completing the Square

Add x^{2} to both sides.

$$-\frac{1}{4}x+\frac{15}{8}+x^{2}=-2x+3$$

Add 2x to both sides.

$$-\frac{1}{4}x+\frac{15}{8}+x^{2}+2x=3$$

Combine -\frac{1}{4}x and 2x to get \frac{7}{4}x.

$$\frac{7}{4}x+\frac{15}{8}+x^{2}=3$$

Subtract \frac{15}{8} from both sides.

$$\frac{7}{4}x+x^{2}=3-\frac{15}{8}$$

Subtract \frac{15}{8} from 3 to get \frac{9}{8}.

$$\frac{7}{4}x+x^{2}=\frac{9}{8}$$

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.

$$x^{2}+\frac{7}{4}x=\frac{9}{8}$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\frac{9}{8}+\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{9}{8}+\frac{49}{64}$$

Add \frac{9}{8} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{121}{64}$$

Factor x^{2}+\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{121}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Simplify.

$$x+\frac{7}{8}=\frac{11}{8} x+\frac{7}{8}=-\frac{11}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$x=\frac{1}{2} x=-\frac{9}{4}$$