Solve left(x+101right)^2+10(x+101)+9=0

$${ \left(x+101 \right) }^{ 2 } +10(x+101)+9=0$$

Solve for x

$$x=-110$$

$$x=-102$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+101\right)^{2}.

$$x^{2}+202x+10201+10\left(x+101\right)+9=0$$

Use the distributive property to multiply 10 by x+101.

$$x^{2}+202x+10201+10x+1010+9=0$$

Combine 202x and 10x to get 212x.

$$x^{2}+212x+10201+1010+9=0$$

Add 10201 and 1010 to get 11211.

$$x^{2}+212x+11211+9=0$$

Add 11211 and 9 to get 11220.

$$x^{2}+212x+11220=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 212 for b, and 11220 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-212±\sqrt{212^{2}-4\times 11220}}{2}$$

Square 212.

$$x=\frac{-212±\sqrt{44944-4\times 11220}}{2}$$

Multiply -4 times 11220.

$$x=\frac{-212±\sqrt{44944-44880}}{2}$$

Add 44944 to -44880.

$$x=\frac{-212±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-212±8}{2}$$

Now solve the equation x=\frac{-212±8}{2} when ± is plus. Add -212 to 8.

$$x=-\frac{204}{2}$$

Divide -204 by 2.

$$x=-102$$

Now solve the equation x=\frac{-212±8}{2} when ± is minus. Subtract 8 from -212.

$$x=-\frac{220}{2}$$

Divide -220 by 2.

$$x=-110$$

The equation is now solved.

$$x=-102 x=-110$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+101\right)^{2}.

$$x^{2}+202x+10201+10\left(x+101\right)+9=0$$

Use the distributive property to multiply 10 by x+101.

$$x^{2}+202x+10201+10x+1010+9=0$$

Combine 202x and 10x to get 212x.

$$x^{2}+212x+10201+1010+9=0$$

Add 10201 and 1010 to get 11211.

$$x^{2}+212x+11211+9=0$$

Add 11211 and 9 to get 11220.

$$x^{2}+212x+11220=0$$

Subtract 11220 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+212x=-11220$$

Divide 212, the coefficient of the x term, by 2 to get 106. Then add the square of 106 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+212x+106^{2}=-11220+106^{2}$$

Square 106.

$$x^{2}+212x+11236=-11220+11236$$

Add -11220 to 11236.

$$x^{2}+212x+11236=16$$

Factor x^{2}+212x+11236. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+106\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+106\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+106=4 x+106=-4$$

Subtract 106 from both sides of the equation.

$$x=-102 x=-110$$