Solve 3x^2-2x=81-2x-x^2

$$3 x ^ { 2 } - 2 x = 81 - 2 x - x ^ { 2 }$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps Using Difference of Squares

Subtract 81 from both sides.

$$3x^{2}-2x-81=-2x-x^{2}$$

Add 2x to both sides.

$$3x^{2}-2x-81+2x=-x^{2}$$

Combine -2x and 2x to get 0.

$$3x^{2}-81=-x^{2}$$

Add x^{2} to both sides.

$$3x^{2}-81+x^{2}=0$$

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

$$4x^{2}-81=0$$

Consider 4x^{2}-81. Rewrite 4x^{2}-81 as \left(2x\right)^{2}-9^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x-9\right)\left(2x+9\right)=0$$

To find equation solutions, solve 2x-9=0 and 2x+9=0.

$$x=\frac{9}{2} x=-\frac{9}{2}$$

Steps by Finding Square Root

Add 2x to both sides.

$$3x^{2}-2x+2x=81-x^{2}$$

Combine -2x and 2x to get 0.

$$3x^{2}=81-x^{2}$$

Add x^{2} to both sides.

$$3x^{2}+x^{2}=81$$

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

$$4x^{2}=81$$

Divide both sides by 4.

$$x^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$x=\frac{9}{2} x=-\frac{9}{2}$$

Steps Using the Quadratic Formula

Subtract 81 from both sides.

$$3x^{2}-2x-81=-2x-x^{2}$$

Add 2x to both sides.

$$3x^{2}-2x-81+2x=-x^{2}$$

Combine -2x and 2x to get 0.

$$3x^{2}-81=-x^{2}$$

Add x^{2} to both sides.

$$3x^{2}-81+x^{2}=0$$

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

$$4x^{2}-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 0 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 4\left(-81\right)}}{2\times 4}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4\left(-81\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\left(-81\right)}}{2\times 4}$$

Multiply -16 times -81.

$$x=\frac{0±\sqrt{1296}}{2\times 4}$$

Take the square root of 1296.

$$x=\frac{0±36}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±36}{8}$$

Now solve the equation x=\frac{0±36}{8} when ± is plus. Reduce the fraction \frac{36}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{9}{2}$$

Now solve the equation x=\frac{0±36}{8} when ± is minus. Reduce the fraction \frac{-36}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{9}{2}$$

The equation is now solved.

$$x=\frac{9}{2} x=-\frac{9}{2}$$