Solve 3m^2+9=11-23m

$$3 m ^ { 2 } + 9 = 11 - 23 m$$

Solve for m

$$m=\frac{\sqrt{553}-23}{6}\approx 0.085992005$$

$$m=\frac{-\sqrt{553}-23}{6}\approx -7.752658672$$

Steps Using the Quadratic Formula

Subtract 11 from both sides.

$$3m^{2}+9-11=-23m$$

Subtract 11 from 9 to get -2.

$$3m^{2}-2=-23m$$

Add 23m to both sides.

$$3m^{2}-2+23m=0$$

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.

$$3m^{2}+23m-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 23 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-23±\sqrt{23^{2}-4\times 3\left(-2\right)}}{2\times 3}$$

Square 23.

$$m=\frac{-23±\sqrt{529-4\times 3\left(-2\right)}}{2\times 3}$$

Multiply -4 times 3.

$$m=\frac{-23±\sqrt{529-12\left(-2\right)}}{2\times 3}$$

Multiply -12 times -2.

$$m=\frac{-23±\sqrt{529+24}}{2\times 3}$$

Add 529 to 24.

$$m=\frac{-23±\sqrt{553}}{2\times 3}$$

Multiply 2 times 3.

$$m=\frac{-23±\sqrt{553}}{6}$$

Now solve the equation m=\frac{-23±\sqrt{553}}{6} when ± is plus. Add -23 to \sqrt{553}.

$$m=\frac{\sqrt{553}-23}{6}$$

Now solve the equation m=\frac{-23±\sqrt{553}}{6} when ± is minus. Subtract \sqrt{553} from -23.

$$m=\frac{-\sqrt{553}-23}{6}$$

The equation is now solved.

$$m=\frac{\sqrt{553}-23}{6} m=\frac{-\sqrt{553}-23}{6}$$

Steps for Completing the Square

Add 23m to both sides.

$$3m^{2}+9+23m=11$$

Subtract 9 from both sides.

$$3m^{2}+23m=11-9$$

Subtract 9 from 11 to get 2.

$$3m^{2}+23m=2$$

Divide both sides by 3.

$$\frac{3m^{2}+23m}{3}=\frac{2}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$m^{2}+\frac{23}{3}m=\frac{2}{3}$$

Divide \frac{23}{3}, the coefficient of the x term, by 2 to get \frac{23}{6}. Then add the square of \frac{23}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{23}{3}m+\left(\frac{23}{6}\right)^{2}=\frac{2}{3}+\left(\frac{23}{6}\right)^{2}$$

Square \frac{23}{6} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{23}{3}m+\frac{529}{36}=\frac{2}{3}+\frac{529}{36}$$

Add \frac{2}{3} to \frac{529}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{23}{3}m+\frac{529}{36}=\frac{553}{36}$$

Factor m^{2}+\frac{23}{3}m+\frac{529}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{23}{6}\right)^{2}=\frac{553}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{23}{6}\right)^{2}}=\sqrt{\frac{553}{36}}$$

Simplify.

$$m+\frac{23}{6}=\frac{\sqrt{553}}{6} m+\frac{23}{6}=-\frac{\sqrt{553}}{6}$$

Subtract \frac{23}{6} from both sides of the equation.

$$m=\frac{\sqrt{553}-23}{6} m=\frac{-\sqrt{553}-23}{6}$$