Solve 5/4x=x^2-2x-3

$$\frac { 5 } { 4 } x = x ^ { 2 } - 2 x - 3$$

Solve for x

$$x=4$$

$$x=-\frac{3}{4}=-0.75$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$\frac{5}{4}x-x^{2}=-2x-3$$

Add 2x to both sides.

$$\frac{5}{4}x-x^{2}+2x=-3$$

Combine \frac{5}{4}x and 2x to get \frac{13}{4}x.

$$\frac{13}{4}x-x^{2}=-3$$

Add 3 to both sides.

$$\frac{13}{4}x-x^{2}+3=0$$

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.

$$-x^{2}+\frac{13}{4}x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{13}{4} for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{13}{4}±\sqrt{\left(\frac{13}{4}\right)^{2}-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Square \frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{13}{4}±\sqrt{\frac{169}{16}-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{13}{4}±\sqrt{\frac{169}{16}+4\times 3}}{2\left(-1\right)}$$

Multiply 4 times 3.

$$x=\frac{-\frac{13}{4}±\sqrt{\frac{169}{16}+12}}{2\left(-1\right)}$$

Add \frac{169}{16} to 12.

$$x=\frac{-\frac{13}{4}±\sqrt{\frac{361}{16}}}{2\left(-1\right)}$$

Take the square root of \frac{361}{16}.

$$x=\frac{-\frac{13}{4}±\frac{19}{4}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{13}{4}±\frac{19}{4}}{-2}$$

Now solve the equation x=\frac{-\frac{13}{4}±\frac{19}{4}}{-2} when ± is plus. Add -\frac{13}{4} to \frac{19}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{3}{2}}{-2}$$

Divide \frac{3}{2} by -2.

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

Now solve the equation x=\frac{-\frac{13}{4}±\frac{19}{4}}{-2} when ± is minus. Subtract \frac{19}{4} from -\frac{13}{4} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -8 by -2.

$$x=4$$

The equation is now solved.

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

Steps for Completing the Square

Subtract x^{2} from both sides.

$$\frac{5}{4}x-x^{2}=-2x-3$$

Add 2x to both sides.

$$\frac{5}{4}x-x^{2}+2x=-3$$

Combine \frac{5}{4}x and 2x to get \frac{13}{4}x.

$$\frac{13}{4}x-x^{2}=-3$$

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.

$$-x^{2}+\frac{13}{4}x=-3$$

Divide both sides by -1.

$$\frac{-x^{2}+\frac{13}{4}x}{-1}=-\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{\frac{13}{4}}{-1}x=-\frac{3}{-1}$$

Divide \frac{13}{4} by -1.

$$x^{2}-\frac{13}{4}x=-\frac{3}{-1}$$

Divide -3 by -1.

$$x^{2}-\frac{13}{4}x=3$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{4}x+\left(-\frac{13}{8}\right)^{2}=3+\left(-\frac{13}{8}\right)^{2}$$

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=3+\frac{169}{64}$$

Add 3 to \frac{169}{64}.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{361}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{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}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{361}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{361}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{19}{8} x-\frac{13}{8}=-\frac{19}{8}$$

Add \frac{13}{8} to both sides of the equation.

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