Solve x^2+16x+64=900

$$x ^ { 2 } + 16 x + 64 = 900$$

Solve for x

$$x=-38$$

$$x=22$$

Steps Using Factoring

Subtract 900 from both sides.

$$x^{2}+16x+64-900=0$$

Subtract 900 from 64 to get -836.

$$x^{2}+16x-836=0$$

To solve the equation, factor x^{2}+16x-836 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.

$$a+b=16 ab=-836$$

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 -836.

$$-1,836 -2,418 -4,209 -11,76 -19,44 -22,38$$

Calculate the sum for each pair.

$$-1+836=835 -2+418=416 -4+209=205 -11+76=65 -19+44=25 -22+38=16$$

The solution is the pair that gives sum 16.

$$a=-22 b=38$$

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

$$\left(x-22\right)\left(x+38\right)$$

To find equation solutions, solve x-22=0 and x+38=0.

$$x=22 x=-38$$

Steps Using Factoring By Grouping

Subtract 900 from both sides.

$$x^{2}+16x+64-900=0$$

Subtract 900 from 64 to get -836.

$$x^{2}+16x-836=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-836. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\left(-836\right)=-836$$

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 -836.

$$-1,836 -2,418 -4,209 -11,76 -19,44 -22,38$$

Calculate the sum for each pair.

$$-1+836=835 -2+418=416 -4+209=205 -11+76=65 -19+44=25 -22+38=16$$

The solution is the pair that gives sum 16.

$$a=-22 b=38$$

Rewrite x^{2}+16x-836 as \left(x^{2}-22x\right)+\left(38x-836\right).

$$\left(x^{2}-22x\right)+\left(38x-836\right)$$

Factor out x in the first and 38 in the second group.

$$x\left(x-22\right)+38\left(x-22\right)$$

Factor out common term x-22 by using distributive property.

$$\left(x-22\right)\left(x+38\right)$$

To find equation solutions, solve x-22=0 and x+38=0.

$$x=22 x=-38$$

Steps Using the Quadratic Formula

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.

$$x^{2}+16x+64=900$$

Subtract 900 from both sides of the equation.

$$x^{2}+16x+64-900=900-900$$

Subtracting 900 from itself leaves 0.

$$x^{2}+16x+64-900=0$$

Subtract 900 from 64.

$$x^{2}+16x-836=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and -836 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-836\right)}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-836\right)}}{2}$$

Multiply -4 times -836.

$$x=\frac{-16±\sqrt{256+3344}}{2}$$

Add 256 to 3344.

$$x=\frac{-16±\sqrt{3600}}{2}$$

Take the square root of 3600.

$$x=\frac{-16±60}{2}$$

Now solve the equation x=\frac{-16±60}{2} when ± is plus. Add -16 to 60.

$$x=\frac{44}{2}$$

Divide 44 by 2.

$$x=22$$

Now solve the equation x=\frac{-16±60}{2} when ± is minus. Subtract 60 from -16.

$$x=-\frac{76}{2}$$

Divide -76 by 2.

$$x=-38$$

The equation is now solved.

$$x=22 x=-38$$

Steps for Completing the Square

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+8\right)^{2}=900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{900}$$

Simplify.

$$x+8=30 x+8=-30$$

Subtract 8 from both sides of the equation.

$$x=22 x=-38$$