Solve (x+3)^2+(x-3)^2=(x+3)(x-3)+31

$$( x + 3 ) ^ { 2 } + ( x - 3 ) ^ { 2 } = ( x + 3 ) ( x - 3 ) + 31$$

Solve for x

$$x=-2$$

$$x=2$$

Steps Using Difference of Squares

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+\left(x-3\right)^{2}=\left(x+3\right)\left(x-3\right)+31$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}+6x+9+x^{2}-6x+9=\left(x+3\right)\left(x-3\right)+31$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+6x+9-6x+9=\left(x+3\right)\left(x-3\right)+31$$

Combine 6x and -6x to get 0.

$$2x^{2}+9+9=\left(x+3\right)\left(x-3\right)+31$$

Add 9 and 9 to get 18.

$$2x^{2}+18=\left(x+3\right)\left(x-3\right)+31$$

Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$2x^{2}+18=x^{2}-9+31$$

Add -9 and 31 to get 22.

$$2x^{2}+18=x^{2}+22$$

Subtract x^{2} from both sides.

$$2x^{2}+18-x^{2}=22$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+18=22$$

Subtract 22 from both sides.

$$x^{2}+18-22=0$$

Subtract 22 from 18 to get -4.

$$x^{2}-4=0$$

Consider x^{2}-4. Rewrite x^{2}-4 as x^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-2\right)\left(x+2\right)=0$$

To find equation solutions, solve x-2=0 and x+2=0.

$$x=2 x=-2$$

Steps by Finding Square Root

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+\left(x-3\right)^{2}=\left(x+3\right)\left(x-3\right)+31$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}+6x+9+x^{2}-6x+9=\left(x+3\right)\left(x-3\right)+31$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+6x+9-6x+9=\left(x+3\right)\left(x-3\right)+31$$

Combine 6x and -6x to get 0.

$$2x^{2}+9+9=\left(x+3\right)\left(x-3\right)+31$$

Add 9 and 9 to get 18.

$$2x^{2}+18=\left(x+3\right)\left(x-3\right)+31$$

Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$2x^{2}+18=x^{2}-9+31$$

Add -9 and 31 to get 22.

$$2x^{2}+18=x^{2}+22$$

Subtract x^{2} from both sides.

$$2x^{2}+18-x^{2}=22$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+18=22$$

Subtract 18 from both sides.

$$x^{2}=22-18$$

Subtract 18 from 22 to get 4.

$$x^{2}=4$$

Take the square root of both sides of the equation.

$$x=2 x=-2$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+\left(x-3\right)^{2}=\left(x+3\right)\left(x-3\right)+31$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}+6x+9+x^{2}-6x+9=\left(x+3\right)\left(x-3\right)+31$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+6x+9-6x+9=\left(x+3\right)\left(x-3\right)+31$$

Combine 6x and -6x to get 0.

$$2x^{2}+9+9=\left(x+3\right)\left(x-3\right)+31$$

Add 9 and 9 to get 18.

$$2x^{2}+18=\left(x+3\right)\left(x-3\right)+31$$

Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$2x^{2}+18=x^{2}-9+31$$

Add -9 and 31 to get 22.

$$2x^{2}+18=x^{2}+22$$

Subtract x^{2} from both sides.

$$2x^{2}+18-x^{2}=22$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+18=22$$

Subtract 22 from both sides.

$$x^{2}+18-22=0$$

Subtract 22 from 18 to get -4.

$$x^{2}-4=0$$

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-4\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$x=\frac{0±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{0±4}{2}$$

Now solve the equation x=\frac{0±4}{2} when ± is plus. Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{0±4}{2} when ± is minus. Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=2 x=-2$$