Solve 8m-9m^2=16m^2-8m^2+4m-8m^2

$$8 m - 9 m ^ { 2 } = 16 m ^ { 2 } - 8 m ^ { 2 } + 4 m - 8 m ^ { 2 }$$

Solve for m

$$m=\frac{4}{9}\approx 0.444444444$$

$$m=0$$

Steps Using Factoring

Combine 16m^{2} and -8m^{2} to get 8m^{2}.

$$8m-9m^{2}=8m^{2}+4m-8m^{2}$$

Combine 8m^{2} and -8m^{2} to get 0.

$$8m-9m^{2}=4m$$

Subtract 4m from both sides.

$$8m-9m^{2}-4m=0$$

Combine 8m and -4m to get 4m.

$$4m-9m^{2}=0$$

Factor out m.

$$m\left(4-9m\right)=0$$

To find equation solutions, solve m=0 and 4-9m=0.

$$m=0 m=\frac{4}{9}$$

Steps Using the Quadratic Formula

Combine 16m^{2} and -8m^{2} to get 8m^{2}.

$$8m-9m^{2}=8m^{2}+4m-8m^{2}$$

Combine 8m^{2} and -8m^{2} to get 0.

$$8m-9m^{2}=4m$$

Subtract 4m from both sides.

$$8m-9m^{2}-4m=0$$

Combine 8m and -4m to get 4m.

$$4m-9m^{2}=0$$

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.

$$-9m^{2}+4m=0$$

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

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

Take the square root of 4^{2}.

$$m=\frac{-4±4}{2\left(-9\right)}$$

Multiply 2 times -9.

$$m=\frac{-4±4}{-18}$$

Now solve the equation m=\frac{-4±4}{-18} when ± is plus. Add -4 to 4.

$$m=\frac{0}{-18}$$

Divide 0 by -18.

$$m=0$$

Now solve the equation m=\frac{-4±4}{-18} when ± is minus. Subtract 4 from -4.

$$m=-\frac{8}{-18}$$

Reduce the fraction \frac{-8}{-18} to lowest terms by extracting and canceling out 2.

$$m=\frac{4}{9}$$

The equation is now solved.

$$m=0 m=\frac{4}{9}$$

Steps for Completing the Square

Combine 16m^{2} and -8m^{2} to get 8m^{2}.

$$8m-9m^{2}=8m^{2}+4m-8m^{2}$$

Combine 8m^{2} and -8m^{2} to get 0.

$$8m-9m^{2}=4m$$

Subtract 4m from both sides.

$$8m-9m^{2}-4m=0$$

Combine 8m and -4m to get 4m.

$$4m-9m^{2}=0$$

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.

$$-9m^{2}+4m=0$$

Divide both sides by -9.

$$\frac{-9m^{2}+4m}{-9}=\frac{0}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$m^{2}+\frac{4}{-9}m=\frac{0}{-9}$$

Divide 4 by -9.

$$m^{2}-\frac{4}{9}m=\frac{0}{-9}$$

Divide 0 by -9.

$$m^{2}-\frac{4}{9}m=0$$

Divide -\frac{4}{9}, the coefficient of the x term, by 2 to get -\frac{2}{9}. Then add the square of -\frac{2}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{4}{9}m+\left(-\frac{2}{9}\right)^{2}=\left(-\frac{2}{9}\right)^{2}$$

Square -\frac{2}{9} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{4}{9}m+\frac{4}{81}=\frac{4}{81}$$

Factor m^{2}-\frac{4}{9}m+\frac{4}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{2}{9}\right)^{2}=\frac{4}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{2}{9}\right)^{2}}=\sqrt{\frac{4}{81}}$$

Simplify.

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

Add \frac{2}{9} to both sides of the equation.

$$m=\frac{4}{9} m=0$$