Solve -15m^2/16-15/8m^2+45/16=9/16m^2-frac{27}{8}m+frac{45}{16}

$$- \frac { 15 m ^ { 2 } } { 16 } - \frac { 15 } { 8 } m ^ { 2 } + \frac { 45 } { 16 } = \frac { 9 } { 16 } m ^ { 2 } - \frac { 27 } { 8 } m + \frac { 45 } { 16 }$$

Solve for m

$$m=1$$

$$m=0$$

Steps Using Factoring

Multiply both sides of the equation by 16, the least common multiple of 16,8.

$$-15m^{2}-30m^{2}+45=9m^{2}-54m+45$$

Combine -15m^{2} and -30m^{2} to get -45m^{2}.

$$-45m^{2}+45=9m^{2}-54m+45$$

Subtract 9m^{2} from both sides.

$$-45m^{2}+45-9m^{2}=-54m+45$$

Combine -45m^{2} and -9m^{2} to get -54m^{2}.

$$-54m^{2}+45=-54m+45$$

Add 54m to both sides.

$$-54m^{2}+45+54m=45$$

Subtract 45 from both sides.

$$-54m^{2}+45+54m-45=0$$

Subtract 45 from 45 to get 0.

$$-54m^{2}+54m=0$$

Factor out m.

$$m\left(-54m+54\right)=0$$

To find equation solutions, solve m=0 and -54m+54=0.

$$m=0 m=1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 16, the least common multiple of 16,8.

$$-15m^{2}-30m^{2}+45=9m^{2}-54m+45$$

Combine -15m^{2} and -30m^{2} to get -45m^{2}.

$$-45m^{2}+45=9m^{2}-54m+45$$

Subtract 9m^{2} from both sides.

$$-45m^{2}+45-9m^{2}=-54m+45$$

Combine -45m^{2} and -9m^{2} to get -54m^{2}.

$$-54m^{2}+45=-54m+45$$

Add 54m to both sides.

$$-54m^{2}+45+54m=45$$

Subtract 45 from both sides.

$$-54m^{2}+45+54m-45=0$$

Subtract 45 from 45 to get 0.

$$-54m^{2}+54m=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -54 for a, 54 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-54±\sqrt{54^{2}}}{2\left(-54\right)}$$

Take the square root of 54^{2}.

$$m=\frac{-54±54}{2\left(-54\right)}$$

Multiply 2 times -54.

$$m=\frac{-54±54}{-108}$$

Now solve the equation m=\frac{-54±54}{-108} when ± is plus. Add -54 to 54.

$$m=\frac{0}{-108}$$

Divide 0 by -108.

$$m=0$$

Now solve the equation m=\frac{-54±54}{-108} when ± is minus. Subtract 54 from -54.

$$m=-\frac{108}{-108}$$

Divide -108 by -108.

$$m=1$$

The equation is now solved.

$$m=0 m=1$$

Steps for Completing the Square

Multiply both sides of the equation by 16, the least common multiple of 16,8.

$$-15m^{2}-30m^{2}+45=9m^{2}-54m+45$$

Combine -15m^{2} and -30m^{2} to get -45m^{2}.

$$-45m^{2}+45=9m^{2}-54m+45$$

Subtract 9m^{2} from both sides.

$$-45m^{2}+45-9m^{2}=-54m+45$$

Combine -45m^{2} and -9m^{2} to get -54m^{2}.

$$-54m^{2}+45=-54m+45$$

Add 54m to both sides.

$$-54m^{2}+45+54m=45$$

Subtract 45 from both sides.

$$-54m^{2}+54m=45-45$$

Subtract 45 from 45 to get 0.

$$-54m^{2}+54m=0$$

Divide both sides by -54.

$$\frac{-54m^{2}+54m}{-54}=\frac{0}{-54}$$

Dividing by -54 undoes the multiplication by -54.

$$m^{2}+\frac{54}{-54}m=\frac{0}{-54}$$

Divide 54 by -54.

$$m^{2}-m=\frac{0}{-54}$$

Divide 0 by -54.

$$m^{2}-m=0$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-m+\left(-\frac{1}{2}\right)^{2}=\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-m+\frac{1}{4}=\frac{1}{4}$$

Factor m^{2}-m+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{1}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$m-\frac{1}{2}=\frac{1}{2} m-\frac{1}{2}=-\frac{1}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$m=1 m=0$$