Solve 2(x^2-9)=90-x^2

$$2 ( x ^ { 2 } - 9 ) = 90 - x ^ { 2 }$$

Solve for x

$$x=6$$

$$x=-6$$

Steps Using Difference of Squares

Use the distributive property to multiply 2 by x^{2}-9.

$$2x^{2}-18=90-x^{2}$$

Subtract 90 from both sides.

$$2x^{2}-18-90=-x^{2}$$

Subtract 90 from -18 to get -108.

$$2x^{2}-108=-x^{2}$$

Add x^{2} to both sides.

$$2x^{2}-108+x^{2}=0$$

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

$$3x^{2}-108=0$$

Divide both sides by 3.

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

Consider x^{2}-36. Rewrite x^{2}-36 as x^{2}-6^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-6\right)\left(x+6\right)=0$$

To find equation solutions, solve x-6=0 and x+6=0.

$$x=6 x=-6$$

Steps by Finding Square Root

Use the distributive property to multiply 2 by x^{2}-9.

$$2x^{2}-18=90-x^{2}$$

Add x^{2} to both sides.

$$2x^{2}-18+x^{2}=90$$

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

$$3x^{2}-18=90$$

Add 18 to both sides.

$$3x^{2}=90+18$$

Add 90 and 18 to get 108.

$$3x^{2}=108$$

Divide both sides by 3.

$$x^{2}=\frac{108}{3}$$

Divide 108 by 3 to get 36.

$$x^{2}=36$$

Take the square root of both sides of the equation.

$$x=6 x=-6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by x^{2}-9.

$$2x^{2}-18=90-x^{2}$$

Subtract 90 from both sides.

$$2x^{2}-18-90=-x^{2}$$

Subtract 90 from -18 to get -108.

$$2x^{2}-108=-x^{2}$$

Add x^{2} to both sides.

$$2x^{2}-108+x^{2}=0$$

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

$$3x^{2}-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-108\right)}}{2\times 3}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\left(-108\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{0±\sqrt{-12\left(-108\right)}}{2\times 3}$$

Multiply -12 times -108.

$$x=\frac{0±\sqrt{1296}}{2\times 3}$$

Take the square root of 1296.

$$x=\frac{0±36}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{0±36}{6}$$

Now solve the equation x=\frac{0±36}{6} when ± is plus. Divide 36 by 6.

$$x=6$$

Now solve the equation x=\frac{0±36}{6} when ± is minus. Divide -36 by 6.

$$x=-6$$

The equation is now solved.

$$x=6 x=-6$$