Solve (2x+1)^2-(x+2)^2=9

$$( 2 x + 1 ) ^ { 2 } - ( x + 2 ) ^ { 2 } = 9$$

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(2x+1\right)^{2}.

$$4x^{2}+4x+1-\left(x+2\right)^{2}=9$$

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

$$4x^{2}+4x+1-\left(x^{2}+4x+4\right)=9$$

To find the opposite of x^{2}+4x+4, find the opposite of each term.

$$4x^{2}+4x+1-x^{2}-4x-4=9$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+4x+1-4x-4=9$$

Combine 4x and -4x to get 0.

$$3x^{2}+1-4=9$$

Subtract 4 from 1 to get -3.

$$3x^{2}-3=9$$

Subtract 9 from both sides.

$$3x^{2}-3-9=0$$

Subtract 9 from -3 to get -12.

$$3x^{2}-12=0$$

Divide both sides by 3.

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

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(2x+1\right)^{2}.

$$4x^{2}+4x+1-\left(x+2\right)^{2}=9$$

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

$$4x^{2}+4x+1-\left(x^{2}+4x+4\right)=9$$

To find the opposite of x^{2}+4x+4, find the opposite of each term.

$$4x^{2}+4x+1-x^{2}-4x-4=9$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+4x+1-4x-4=9$$

Combine 4x and -4x to get 0.

$$3x^{2}+1-4=9$$

Subtract 4 from 1 to get -3.

$$3x^{2}-3=9$$

Add 3 to both sides.

$$3x^{2}=9+3$$

Add 9 and 3 to get 12.

$$3x^{2}=12$$

Divide both sides by 3.

$$x^{2}=\frac{12}{3}$$

Divide 12 by 3 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(2x+1\right)^{2}.

$$4x^{2}+4x+1-\left(x+2\right)^{2}=9$$

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

$$4x^{2}+4x+1-\left(x^{2}+4x+4\right)=9$$

To find the opposite of x^{2}+4x+4, find the opposite of each term.

$$4x^{2}+4x+1-x^{2}-4x-4=9$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+4x+1-4x-4=9$$

Combine 4x and -4x to get 0.

$$3x^{2}+1-4=9$$

Subtract 4 from 1 to get -3.

$$3x^{2}-3=9$$

Subtract 9 from both sides.

$$3x^{2}-3-9=0$$

Subtract 9 from -3 to get -12.

$$3x^{2}-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-12\right)}}{2\times 3}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\left(-12\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{0±\sqrt{-12\left(-12\right)}}{2\times 3}$$

Multiply -12 times -12.

$$x=\frac{0±\sqrt{144}}{2\times 3}$$

Take the square root of 144.

$$x=\frac{0±12}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{0±12}{6}$$

Now solve the equation x=\frac{0±12}{6} when ± is plus. Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{0±12}{6} when ± is minus. Divide -12 by 6.

$$x=-2$$

The equation is now solved.

$$x=2 x=-2$$