Solve x^2+14x+81=225-49

$${ x }^{ 2 } +14x+81 = 225-49$$

Solve for x

$$x=-19$$

$$x=5$$

Steps Using Factoring

Subtract 49 from 225 to get 176.

$$x^{2}+14x+81=176$$

Subtract 176 from both sides.

$$x^{2}+14x+81-176=0$$

Subtract 176 from 81 to get -95.

$$x^{2}+14x-95=0$$

To solve the equation, factor x^{2}+14x-95 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=14 ab=-95$$

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 -95.

$$-1,95 -5,19$$

Calculate the sum for each pair.

$$-1+95=94 -5+19=14$$

The solution is the pair that gives sum 14.

$$a=-5 b=19$$

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

$$\left(x-5\right)\left(x+19\right)$$

To find equation solutions, solve x-5=0 and x+19=0.

$$x=5 x=-19$$

Steps Using Factoring By Grouping

Subtract 49 from 225 to get 176.

$$x^{2}+14x+81=176$$

Subtract 176 from both sides.

$$x^{2}+14x+81-176=0$$

Subtract 176 from 81 to get -95.

$$x^{2}+14x-95=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-95. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\left(-95\right)=-95$$

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 -95.

$$-1,95 -5,19$$

Calculate the sum for each pair.

$$-1+95=94 -5+19=14$$

The solution is the pair that gives sum 14.

$$a=-5 b=19$$

Rewrite x^{2}+14x-95 as \left(x^{2}-5x\right)+\left(19x-95\right).

$$\left(x^{2}-5x\right)+\left(19x-95\right)$$

Factor out x in the first and 19 in the second group.

$$x\left(x-5\right)+19\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(x+19\right)$$

To find equation solutions, solve x-5=0 and x+19=0.

$$x=5 x=-19$$

Steps Using the Quadratic Formula

Subtract 49 from 225 to get 176.

$$x^{2}+14x+81=176$$

Subtract 176 from both sides.

$$x^{2}+14x+81-176=0$$

Subtract 176 from 81 to get -95.

$$x^{2}+14x-95=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -95 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-95\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-95\right)}}{2}$$

Multiply -4 times -95.

$$x=\frac{-14±\sqrt{196+380}}{2}$$

Add 196 to 380.

$$x=\frac{-14±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-14±24}{2}$$

Now solve the equation x=\frac{-14±24}{2} when ± is plus. Add -14 to 24.

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

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{-14±24}{2} when ± is minus. Subtract 24 from -14.

$$x=-\frac{38}{2}$$

Divide -38 by 2.

$$x=-19$$

The equation is now solved.

$$x=5 x=-19$$

Steps for Completing the Square

Subtract 49 from 225 to get 176.

$$x^{2}+14x+81=176$$

Subtract 81 from both sides.

$$x^{2}+14x=176-81$$

Subtract 81 from 176 to get 95.

$$x^{2}+14x=95$$

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.

$$x^{2}+14x+7^{2}=95+7^{2}$$

Square 7.

$$x^{2}+14x+49=95+49$$

Add 95 to 49.

$$x^{2}+14x+49=144$$

Factor x^{2}+14x+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}.

$$\left(x+7\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{144}$$

Simplify.

$$x+7=12 x+7=-12$$

Subtract 7 from both sides of the equation.

$$x=5 x=-19$$