Solve 5m/4+76=m^2

$$\frac { 5 m } { 4 } + 76 = m ^ { 2 }$$

Solve for m

$$m = \frac{\sqrt{4889} + 5}{8} \approx 9.365173053$$

$$m=\frac{5-\sqrt{4889}}{8}\approx -8.115173053$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$5m+304=4m^{2}$$

Subtract 4m^{2} from both sides.

$$5m+304-4m^{2}=0$$

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.

$$-4m^{2}+5m+304=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 5 for b, and 304 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-5±\sqrt{5^{2}-4\left(-4\right)\times 304}}{2\left(-4\right)}$$

Square 5.

$$m=\frac{-5±\sqrt{25-4\left(-4\right)\times 304}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$m=\frac{-5±\sqrt{25+16\times 304}}{2\left(-4\right)}$$

Multiply 16 times 304.

$$m=\frac{-5±\sqrt{25+4864}}{2\left(-4\right)}$$

Add 25 to 4864.

$$m=\frac{-5±\sqrt{4889}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$m=\frac{-5±\sqrt{4889}}{-8}$$

Now solve the equation m=\frac{-5±\sqrt{4889}}{-8} when ± is plus. Add -5 to \sqrt{4889}.

$$m=\frac{\sqrt{4889}-5}{-8}$$

Divide -5+\sqrt{4889} by -8.

$$m=\frac{5-\sqrt{4889}}{8}$$

Now solve the equation m=\frac{-5±\sqrt{4889}}{-8} when ± is minus. Subtract \sqrt{4889} from -5.

$$m=\frac{-\sqrt{4889}-5}{-8}$$

Divide -5-\sqrt{4889} by -8.

$$m=\frac{\sqrt{4889}+5}{8}$$

The equation is now solved.

$$m=\frac{5-\sqrt{4889}}{8} m=\frac{\sqrt{4889}+5}{8}$$

Steps for Completing the Square

Multiply both sides of the equation by 4.

$$5m+304=4m^{2}$$

Subtract 4m^{2} from both sides.

$$5m+304-4m^{2}=0$$

Subtract 304 from both sides. Anything subtracted from zero gives its negation.

$$5m-4m^{2}=-304$$

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.

$$-4m^{2}+5m=-304$$

Divide both sides by -4.

$$\frac{-4m^{2}+5m}{-4}=-\frac{304}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$m^{2}+\frac{5}{-4}m=-\frac{304}{-4}$$

Divide 5 by -4.

$$m^{2}-\frac{5}{4}m=-\frac{304}{-4}$$

Divide -304 by -4.

$$m^{2}-\frac{5}{4}m=76$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{5}{4}m+\left(-\frac{5}{8}\right)^{2}=76+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{5}{4}m+\frac{25}{64}=76+\frac{25}{64}$$

Add 76 to \frac{25}{64}.

$$m^{2}-\frac{5}{4}m+\frac{25}{64}=\frac{4889}{64}$$

Factor m^{2}-\frac{5}{4}m+\frac{25}{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}.

$$\left(m-\frac{5}{8}\right)^{2}=\frac{4889}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{8}\right)^{2}}=\sqrt{\frac{4889}{64}}$$

Simplify.

$$m-\frac{5}{8}=\frac{\sqrt{4889}}{8} m-\frac{5}{8}=-\frac{\sqrt{4889}}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$m=\frac{\sqrt{4889}+5}{8} m=\frac{5-\sqrt{4889}}{8}$$