Solve m^2=9m-8

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

Solve for m

$$m=1$$

$$m=8$$

Steps Using Factoring

Subtract 9m from both sides.

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

Add 8 to both sides.

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

To solve the equation, factor m^{2}-9m+8 using formula m^{2}+\left(a+b\right)m+ab=\left(m+a\right)\left(m+b\right). To find a and b, set up a system to be solved.

$$a+b=-9 ab=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -9.

$$a=-8 b=-1$$

Rewrite factored expression \left(m+a\right)\left(m+b\right) using the obtained values.

$$\left(m-8\right)\left(m-1\right)$$

To find equation solutions, solve m-8=0 and m-1=0.

$$m=8 m=1$$

Steps Using Factoring By Grouping

Subtract 9m from both sides.

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

Add 8 to both sides.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm+8. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\times 8=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -9.

$$a=-8 b=-1$$

Rewrite m^{2}-9m+8 as \left(m^{2}-8m\right)+\left(-m+8\right).

$$\left(m^{2}-8m\right)+\left(-m+8\right)$$

Factor out m in the first and -1 in the second group.

$$m\left(m-8\right)-\left(m-8\right)$$

Factor out common term m-8 by using distributive property.

$$\left(m-8\right)\left(m-1\right)$$

To find equation solutions, solve m-8=0 and m-1=0.

$$m=8 m=1$$

Steps Using the Quadratic Formula

Subtract 9m from both sides.

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

Add 8 to both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 8}}{2}$$

Square -9.

$$m=\frac{-\left(-9\right)±\sqrt{81-4\times 8}}{2}$$

Multiply -4 times 8.

$$m=\frac{-\left(-9\right)±\sqrt{81-32}}{2}$$

Add 81 to -32.

$$m=\frac{-\left(-9\right)±\sqrt{49}}{2}$$

Take the square root of 49.

$$m=\frac{-\left(-9\right)±7}{2}$$

The opposite of -9 is 9.

$$m=\frac{9±7}{2}$$

Now solve the equation m=\frac{9±7}{2} when ± is plus. Add 9 to 7.

$$m=\frac{16}{2}$$

Divide 16 by 2.

$$m=8$$

Now solve the equation m=\frac{9±7}{2} when ± is minus. Subtract 7 from 9.

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

Divide 2 by 2.

$$m=1$$

The equation is now solved.

$$m=8 m=1$$

Steps for Completing the Square

Subtract 9m from both sides.

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

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

$$m^{2}-9m+\left(-\frac{9}{2}\right)^{2}=-8+\left(-\frac{9}{2}\right)^{2}$$

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

$$m^{2}-9m+\frac{81}{4}=-8+\frac{81}{4}$$

Add -8 to \frac{81}{4}.

$$m^{2}-9m+\frac{81}{4}=\frac{49}{4}$$

Factor m^{2}-9m+\frac{81}{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}.

$$\left(m-\frac{9}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{9}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$m-\frac{9}{2}=\frac{7}{2} m-\frac{9}{2}=-\frac{7}{2}$$

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

$$m=8 m=1$$