Solve m^2=11m-30

$$m ^ { 2 } = 11 m - 30$$

Solve for m

$$m=5$$

$$m=6$$

Steps Using Factoring

Subtract 11m from both sides.

$$m^{2}-11m=-30$$

Add 30 to both sides.

$$m^{2}-11m+30=0$$

To solve the equation, factor m^{2}-11m+30 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.

$$a+b=-11 ab=30$$

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 30.

$$-1,-30 -2,-15 -3,-10 -5,-6$$

Calculate the sum for each pair.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

The solution is the pair that gives sum -11.

$$a=-6 b=-5$$

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

$$\left(m-6\right)\left(m-5\right)$$

To find equation solutions, solve m-6=0 and m-5=0.

$$m=6 m=5$$

Steps Using Factoring By Grouping

Subtract 11m from both sides.

$$m^{2}-11m=-30$$

Add 30 to both sides.

$$m^{2}-11m+30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm+30. To find a and b, set up a system to be solved.

$$a+b=-11 ab=1\times 30=30$$

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 30.

$$-1,-30 -2,-15 -3,-10 -5,-6$$

Calculate the sum for each pair.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

The solution is the pair that gives sum -11.

$$a=-6 b=-5$$

Rewrite m^{2}-11m+30 as \left(m^{2}-6m\right)+\left(-5m+30\right).

$$\left(m^{2}-6m\right)+\left(-5m+30\right)$$

Factor out m in the first and -5 in the second group.

$$m\left(m-6\right)-5\left(m-6\right)$$

Factor out common term m-6 by using distributive property.

$$\left(m-6\right)\left(m-5\right)$$

To find equation solutions, solve m-6=0 and m-5=0.

$$m=6 m=5$$

Steps Using the Quadratic Formula

Subtract 11m from both sides.

$$m^{2}-11m=-30$$

Add 30 to both sides.

$$m^{2}-11m+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 30}}{2}$$

Square -11.

$$m=\frac{-\left(-11\right)±\sqrt{121-4\times 30}}{2}$$

Multiply -4 times 30.

$$m=\frac{-\left(-11\right)±\sqrt{121-120}}{2}$$

Add 121 to -120.

$$m=\frac{-\left(-11\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$m=\frac{-\left(-11\right)±1}{2}$$

The opposite of -11 is 11.

$$m=\frac{11±1}{2}$$

Now solve the equation m=\frac{11±1}{2} when ± is plus. Add 11 to 1.

$$m=\frac{12}{2}$$

Divide 12 by 2.

$$m=6$$

Now solve the equation m=\frac{11±1}{2} when ± is minus. Subtract 1 from 11.

$$m=\frac{10}{2}$$

Divide 10 by 2.

$$m=5$$

The equation is now solved.

$$m=6 m=5$$

Steps for Completing the Square

Subtract 11m from both sides.

$$m^{2}-11m=-30$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-11m+\left(-\frac{11}{2}\right)^{2}=-30+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-11m+\frac{121}{4}=-30+\frac{121}{4}$$

Add -30 to \frac{121}{4}.

$$m^{2}-11m+\frac{121}{4}=\frac{1}{4}$$

Factor m^{2}-11m+\frac{121}{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}.

$$\left(m-\frac{11}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{11}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$m-\frac{11}{2}=\frac{1}{2} m-\frac{11}{2}=-\frac{1}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$m=6 m=5$$