Solve m^2+1+2m-64=0

$$m ^ { 2 } + 1 + 2 m - 64 = 0$$

Solve for m

$$m=-9$$

$$m=7$$

Steps Using Factoring

Subtract 64 from 1 to get -63.

$$m^{2}-63+2m=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$m^{2}+2m-63=0$$

To solve the equation, factor m^{2}+2m-63 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.

$$a+b=2 ab=-63$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -63.

$$-1,63 -3,21 -7,9$$

Calculate the sum for each pair.

$$-1+63=62 -3+21=18 -7+9=2$$

The solution is the pair that gives sum 2.

$$a=-7 b=9$$

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

$$\left(m-7\right)\left(m+9\right)$$

To find equation solutions, solve m-7=0 and m+9=0.

$$m=7 m=-9$$

Steps Using Factoring By Grouping

Subtract 64 from 1 to get -63.

$$m^{2}-63+2m=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$m^{2}+2m-63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm-63. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-63\right)=-63$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -63.

$$-1,63 -3,21 -7,9$$

Calculate the sum for each pair.

$$-1+63=62 -3+21=18 -7+9=2$$

The solution is the pair that gives sum 2.

$$a=-7 b=9$$

Rewrite m^{2}+2m-63 as \left(m^{2}-7m\right)+\left(9m-63\right).

$$\left(m^{2}-7m\right)+\left(9m-63\right)$$

Factor out m in the first and 9 in the second group.

$$m\left(m-7\right)+9\left(m-7\right)$$

Factor out common term m-7 by using distributive property.

$$\left(m-7\right)\left(m+9\right)$$

To find equation solutions, solve m-7=0 and m+9=0.

$$m=7 m=-9$$

Steps Using the Quadratic Formula

Subtract 64 from 1 to get -63.

$$m^{2}-63+2m=0$$

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.

$$m^{2}+2m-63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-2±\sqrt{2^{2}-4\left(-63\right)}}{2}$$

Square 2.

$$m=\frac{-2±\sqrt{4-4\left(-63\right)}}{2}$$

Multiply -4 times -63.

$$m=\frac{-2±\sqrt{4+252}}{2}$$

Add 4 to 252.

$$m=\frac{-2±\sqrt{256}}{2}$$

Take the square root of 256.

$$m=\frac{-2±16}{2}$$

Now solve the equation m=\frac{-2±16}{2} when ± is plus. Add -2 to 16.

$$m=\frac{14}{2}$$

Divide 14 by 2.

$$m=7$$

Now solve the equation m=\frac{-2±16}{2} when ± is minus. Subtract 16 from -2.

$$m=-\frac{18}{2}$$

Divide -18 by 2.

$$m=-9$$

The equation is now solved.

$$m=7 m=-9$$

Steps for Completing the Square

Subtract 64 from 1 to get -63.

$$m^{2}-63+2m=0$$

Add 63 to both sides. Anything plus zero gives itself.

$$m^{2}+2m=63$$

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

$$m^{2}+2m+1^{2}=63+1^{2}$$

Square 1.

$$m^{2}+2m+1=63+1$$

Add 63 to 1.

$$m^{2}+2m+1=64$$

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

$$\left(m+1\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+1\right)^{2}}=\sqrt{64}$$

Simplify.

$$m+1=8 m+1=-8$$

Subtract 1 from both sides of the equation.

$$m=7 m=-9$$