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

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $81$ from both sides of the equation.

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

Add $x^{2}$ to both sides of the equation.

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

Step 2

Add $3 x^{2}$ and $x^{2}$ .

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

Step 3

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

${(2 x)}^{2} - 81 = 0$

Step 4

Rewrite $81$ as $9^{2}$ .

${(2 x)}^{2} - 9^{2} = 0$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 9$ .

$(2 x + 9) (2 x - 9) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 9 = 0$

$2 x - 9 = 0$

Step 7

Set $2 x + 9$ equal to $0$ and solve for $x$ .

Set $2 x + 9$ equal to $0$ .

$2 x + 9 = 0$

Solve $2 x + 9 = 0$ for $x$ .

Subtract $9$ from both sides of the equation.

$2 x = - 9$

Divide each term in $2 x = - 9$ by $2$ and simplify.

Divide each term in $2 x = - 9$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 8

Set $2 x - 9$ equal to $0$ and solve for $x$ .

Set $2 x - 9$ equal to $0$ .

$2 x - 9 = 0$

Solve $2 x - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

$2 x = 9$

Divide each term in $2 x = 9$ by $2$ and simplify.

Divide each term in $2 x = 9$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 9

The final solution is all the values that make $(2 x + 9) (2 x - 9) = 0$ true.

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

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

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