Solve 1+x+(1+x)x=36

$$1 + x + ( 1 + x ) x = 36$$

Solve for x

$$x=-7$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1+x by x.

$$1+x+x+x^{2}=36$$

Combine x and x to get 2x.

$$1+2x+x^{2}=36$$

Subtract 36 from both sides.

$$1+2x+x^{2}-36=0$$

Subtract 36 from 1 to get -35.

$$-35+2x+x^{2}=0$$

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.

$$x^{2}+2x-35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-35\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-35\right)}}{2}$$

Multiply -4 times -35.

$$x=\frac{-2±\sqrt{4+140}}{2}$$

Add 4 to 140.

$$x=\frac{-2±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-2±12}{2}$$

Now solve the equation x=\frac{-2±12}{2} when ± is plus. Add -2 to 12.

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

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{-2±12}{2} when ± is minus. Subtract 12 from -2.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

The equation is now solved.

$$x=5 x=-7$$

Steps for Completing the Square

Use the distributive property to multiply 1+x by x.

$$1+x+x+x^{2}=36$$

Combine x and x to get 2x.

$$1+2x+x^{2}=36$$

Subtract 1 from both sides.

$$2x+x^{2}=36-1$$

Subtract 1 from 36 to get 35.

$$2x+x^{2}=35$$

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.

$$x^{2}+2x=35$$

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.

$$x^{2}+2x+1^{2}=35+1^{2}$$

Square 1.

$$x^{2}+2x+1=35+1$$

Add 35 to 1.

$$x^{2}+2x+1=36$$

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}.

$$\left(x+1\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+1=6 x+1=-6$$

Subtract 1 from both sides of the equation.

$$x=5 x=-7$$