Solve (x+2)^2/2+x^2-18/3=2x+2

$$\frac { ( x + 2 ) ^ { 2 } } { 2 } + \frac { x ^ { 2 } - 18 } { 3 } = 2 x + 2$$

Solve for x

$$x = \frac{6 \sqrt{5}}{5} \approx 2.683281573$$

$$x = -\frac{6 \sqrt{5}}{5} \approx -2.683281573$$

Steps by Finding Square Root

Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(x+2\right)^{2}+2\left(x^{2}-18\right)=12x+12$$

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

$$3\left(x^{2}+4x+4\right)+2\left(x^{2}-18\right)=12x+12$$

Use the distributive property to multiply 3 by x^{2}+4x+4.

$$3x^{2}+12x+12+2\left(x^{2}-18\right)=12x+12$$

Use the distributive property to multiply 2 by x^{2}-18.

$$3x^{2}+12x+12+2x^{2}-36=12x+12$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$5x^{2}+12x+12-36=12x+12$$

Subtract 36 from 12 to get -24.

$$5x^{2}+12x-24=12x+12$$

Subtract 12x from both sides.

$$5x^{2}+12x-24-12x=12$$

Combine 12x and -12x to get 0.

$$5x^{2}-24=12$$

Add 24 to both sides.

$$5x^{2}=12+24$$

Add 12 and 24 to get 36.

$$5x^{2}=36$$

Divide both sides by 5.

$$x^{2}=\frac{36}{5}$$

Take the square root of both sides of the equation.

$$x=\frac{6\sqrt{5}}{5} x=-\frac{6\sqrt{5}}{5}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(x+2\right)^{2}+2\left(x^{2}-18\right)=12x+12$$

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

$$3\left(x^{2}+4x+4\right)+2\left(x^{2}-18\right)=12x+12$$

Use the distributive property to multiply 3 by x^{2}+4x+4.

$$3x^{2}+12x+12+2\left(x^{2}-18\right)=12x+12$$

Use the distributive property to multiply 2 by x^{2}-18.

$$3x^{2}+12x+12+2x^{2}-36=12x+12$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$5x^{2}+12x+12-36=12x+12$$

Subtract 36 from 12 to get -24.

$$5x^{2}+12x-24=12x+12$$

Subtract 12x from both sides.

$$5x^{2}+12x-24-12x=12$$

Combine 12x and -12x to get 0.

$$5x^{2}-24=12$$

Subtract 12 from both sides.

$$5x^{2}-24-12=0$$

Subtract 12 from -24 to get -36.

$$5x^{2}-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 0 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 5\left(-36\right)}}{2\times 5}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 5\left(-36\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{0±\sqrt{-20\left(-36\right)}}{2\times 5}$$

Multiply -20 times -36.

$$x=\frac{0±\sqrt{720}}{2\times 5}$$

Take the square root of 720.

$$x=\frac{0±12\sqrt{5}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{0±12\sqrt{5}}{10}$$

Now solve the equation x=\frac{0±12\sqrt{5}}{10} when ± is plus.

$$x=\frac{6\sqrt{5}}{5}$$

Now solve the equation x=\frac{0±12\sqrt{5}}{10} when ± is minus.

$$x=-\frac{6\sqrt{5}}{5}$$

The equation is now solved.

$$x=\frac{6\sqrt{5}}{5} x=-\frac{6\sqrt{5}}{5}$$