Solve by Factoring 3x^2-21=4-x^2

$3 x^{2} - 21 = 4 - x^{2}$

Step 1

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

Subtract $4$ from both sides of the equation.

$3 x^{2} - 21 - 4 = - x^{2}$

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

$3 x^{2} - 21 - 4 + x^{2} = 0$

Step 2

Simplify $3 x^{2} - 21 - 4 + x^{2}$ .

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

$4 x^{2} - 21 - 4 = 0$

Subtract $4$ from $- 21$ .

$4 x^{2} - 25 = 0$

Step 3

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

${(2 x)}^{2} - 25 = 0$

Step 4

Rewrite $25$ as $5^{2}$ .

${(2 x)}^{2} - 5^{2} = 0$

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 $b = 5$ .

$(2 x + 5) (2 x - 5) = 0$

Step 6

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

$2 x + 5 = 0$

$2 x - 5 = 0$

Step 7

Set $2 x + 5$ equal to $0$ and solve for $x$ .

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Solve $2 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 8

Set $2 x - 5$ equal to $0$ and solve for $x$ .

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Solve $2 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$2 x = 5$

Divide each term in $2 x = 5$ by $2$ and simplify.

Divide each term in $2 x = 5$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 9

The final solution is all the values that make $(2 x + 5) (2 x - 5) = 0$ true.

$x = - \frac{5}{2}, \frac{5}{2}$

Solve by Factoring 3x^2-21=4-x^2

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