Solve 49-14m-14+m^2+5=0

$$49-14m-14+ { m }^{ 2 } +5 = 0$$

Solve for m

$$m=4$$

$$m=10$$

Steps Using Factoring

Subtract 14 from 49 to get 35.

$$35-14m+m^{2}+5=0$$

Add 35 and 5 to get 40.

$$40-14m+m^{2}=0$$

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

$$m^{2}-14m+40=0$$

To solve the equation, factor m^{2}-14m+40 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.

$$a+b=-14 ab=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Calculate the sum for each pair.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

The solution is the pair that gives sum -14.

$$a=-10 b=-4$$

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

$$\left(m-10\right)\left(m-4\right)$$

To find equation solutions, solve m-10=0 and m-4=0.

$$m=10 m=4$$

Steps Using Factoring By Grouping

Subtract 14 from 49 to get 35.

$$35-14m+m^{2}+5=0$$

Add 35 and 5 to get 40.

$$40-14m+m^{2}=0$$

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

$$m^{2}-14m+40=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm+40. To find a and b, set up a system to be solved.

$$a+b=-14 ab=1\times 40=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Calculate the sum for each pair.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

The solution is the pair that gives sum -14.

$$a=-10 b=-4$$

Rewrite m^{2}-14m+40 as \left(m^{2}-10m\right)+\left(-4m+40\right).

$$\left(m^{2}-10m\right)+\left(-4m+40\right)$$

Factor out m in the first and -4 in the second group.

$$m\left(m-10\right)-4\left(m-10\right)$$

Factor out common term m-10 by using distributive property.

$$\left(m-10\right)\left(m-4\right)$$

To find equation solutions, solve m-10=0 and m-4=0.

$$m=10 m=4$$

Steps Using the Quadratic Formula

Subtract 14 from 49 to get 35.

$$35-14m+m^{2}+5=0$$

Add 35 and 5 to get 40.

$$40-14m+m^{2}=0$$

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.

$$m^{2}-14m+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 40}}{2}$$

Square -14.

$$m=\frac{-\left(-14\right)±\sqrt{196-4\times 40}}{2}$$

Multiply -4 times 40.

$$m=\frac{-\left(-14\right)±\sqrt{196-160}}{2}$$

Add 196 to -160.

$$m=\frac{-\left(-14\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$m=\frac{-\left(-14\right)±6}{2}$$

The opposite of -14 is 14.

$$m=\frac{14±6}{2}$$

Now solve the equation m=\frac{14±6}{2} when ± is plus. Add 14 to 6.

$$m=\frac{20}{2}$$

Divide 20 by 2.

$$m=10$$

Now solve the equation m=\frac{14±6}{2} when ± is minus. Subtract 6 from 14.

$$m=\frac{8}{2}$$

Divide 8 by 2.

$$m=4$$

The equation is now solved.

$$m=10 m=4$$

Steps for Completing the Square

Subtract 14 from 49 to get 35.

$$35-14m+m^{2}+5=0$$

Add 35 and 5 to get 40.

$$40-14m+m^{2}=0$$

Subtract 40 from both sides. Anything subtracted from zero gives its negation.

$$-14m+m^{2}=-40$$

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.

$$m^{2}-14m=-40$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-14m+\left(-7\right)^{2}=-40+\left(-7\right)^{2}$$

Square -7.

$$m^{2}-14m+49=-40+49$$

Add -40 to 49.

$$m^{2}-14m+49=9$$

Factor m^{2}-14m+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(m-7\right)^{2}}=\sqrt{9}$$

Simplify.

$$m-7=3 m-7=-3$$

Add 7 to both sides of the equation.

$$m=10 m=4$$