Solve x^2+2x+11=111

$$x ^ { 2 } + 2 x + 11 = 111$$

Solve for x (complex solution)

$$x=\sqrt{101}-1\approx 9.049875621$$

$$x=-\left(\sqrt{101}+1\right)\approx -11.049875621$$

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.

$$x^{2}+2x+11=111$$

Subtract 111 from both sides of the equation.

$$x^{2}+2x+11-111=111-111$$

Subtracting 111 from itself leaves 0.

$$x^{2}+2x+11-111=0$$

Subtract 111 from 11.

$$x^{2}+2x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-100\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-100\right)}}{2}$$

Multiply -4 times -100.

$$x=\frac{-2±\sqrt{4+400}}{2}$$

Add 4 to 400.

$$x=\frac{-2±\sqrt{404}}{2}$$

Take the square root of 404.

$$x=\frac{-2±2\sqrt{101}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{101}}{2} when ± is plus. Add -2 to 2\sqrt{101}.

$$x=\frac{2\sqrt{101}-2}{2}$$

Divide -2+2\sqrt{101} by 2.

$$x=\sqrt{101}-1$$

Now solve the equation x=\frac{-2±2\sqrt{101}}{2} when ± is minus. Subtract 2\sqrt{101} from -2.

$$x=\frac{-2\sqrt{101}-2}{2}$$

Divide -2-2\sqrt{101} by 2.

$$x=-\sqrt{101}-1$$

The equation is now solved.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$

Steps for Completing the Square

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.

$$x^{2}+2x+11=111$$

Subtract 11 from both sides of the equation.

$$x^{2}+2x+11-11=111-11$$

Subtracting 11 from itself leaves 0.

$$x^{2}+2x=111-11$$

Subtract 11 from 111.

$$x^{2}+2x=100$$

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.

$$x^{2}+2x+1^{2}=100+1^{2}$$

Square 1.

$$x^{2}+2x+1=100+1$$

Add 100 to 1.

$$x^{2}+2x+1=101$$

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

$$\left(x+1\right)^{2}=101$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{101}$$

Simplify.

$$x+1=\sqrt{101} x+1=-\sqrt{101}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$

Solve for x

$$x=\sqrt{101}-1\approx 9.049875621$$

$$x=-\sqrt{101}-1\approx -11.049875621$$

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.

$$x^{2}+2x+11=111$$

Subtract 111 from both sides of the equation.

$$x^{2}+2x+11-111=111-111$$

Subtracting 111 from itself leaves 0.

$$x^{2}+2x+11-111=0$$

Subtract 111 from 11.

$$x^{2}+2x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-100\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-100\right)}}{2}$$

Multiply -4 times -100.

$$x=\frac{-2±\sqrt{4+400}}{2}$$

Add 4 to 400.

$$x=\frac{-2±\sqrt{404}}{2}$$

Take the square root of 404.

$$x=\frac{-2±2\sqrt{101}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{101}}{2} when ± is plus. Add -2 to 2\sqrt{101}.

$$x=\frac{2\sqrt{101}-2}{2}$$

Divide -2+2\sqrt{101} by 2.

$$x=\sqrt{101}-1$$

Now solve the equation x=\frac{-2±2\sqrt{101}}{2} when ± is minus. Subtract 2\sqrt{101} from -2.

$$x=\frac{-2\sqrt{101}-2}{2}$$

Divide -2-2\sqrt{101} by 2.

$$x=-\sqrt{101}-1$$

The equation is now solved.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$

Steps for Completing the Square

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.

$$x^{2}+2x+11=111$$

Subtract 11 from both sides of the equation.

$$x^{2}+2x+11-11=111-11$$

Subtracting 11 from itself leaves 0.

$$x^{2}+2x=111-11$$

Subtract 11 from 111.

$$x^{2}+2x=100$$

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.

$$x^{2}+2x+1^{2}=100+1^{2}$$

Square 1.

$$x^{2}+2x+1=100+1$$

Add 100 to 1.

$$x^{2}+2x+1=101$$

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

$$\left(x+1\right)^{2}=101$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{101}$$

Simplify.

$$x+1=\sqrt{101} x+1=-\sqrt{101}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$