Solve for x 3x^2+2=11-x^2

$3 x^{2} + 2 = 11 - x^{2}$

Step 1

Move all terms containing $x$ to the left side of the equation.

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

$3 x^{2} + 2 + x^{2} = 11$

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

$4 x^{2} + 2 = 11$

Step 2

Move all terms not containing $x$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$4 x^{2} = 11 - 2$

Subtract $2$ from $11$ .

$4 x^{2} = 9$

Step 3

Divide each term in $4 x^{2} = 9$ by $4$ and simplify.

Divide each term in $4 x^{2} = 9$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{9}{4}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{9}{4}$

Step 4

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{9}{4}}$

Step 5

Simplify $\pm \sqrt{\frac{9}{4}}$ .

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{9}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{\sqrt{3^{2}}}{\sqrt{4}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{3}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{3}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{3}{2}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

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

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{3}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{3}{2}, - \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{2}, - \frac{3}{2}$

Decimal Form:

$x = 1.5, - 1.5$

Mixed Number Form:

$x = 1 \frac{1}{2}, - 1 \frac{1}{2}$

Solve for x 3x^2+2=11-x^2

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