Solve the System of Equations y=-3x^2+2 and y=3

$y = - 3 x^{2} + 2$ and $y = 3$

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

Solve $- 3 x^{2} + 2 = 3$ for $x$ .

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

Subtract $2$ from both sides of the equation.

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

Subtract $2$ from $3$ .

$- 3 x^{2} = 1$

Divide each term in $- 3 x^{2} = 1$ by $- 3$ and simplify.

Divide each term in $- 3 x^{2} = 1$ by $- 3$ .

$\frac{- 3 x^{2}}{- 3} = \frac{1}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x^{2}}{- 3} = \frac{1}{- 3}$

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

$x^{2} = \frac{1}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{3}$

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

$x = \pm \sqrt{- \frac{1}{3}}$

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{3}}$

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{3}}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{3}}$

One to any power is one.

$x = \pm i \sqrt{\frac{1}{3}}$

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{3}}$

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{3}}$

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = \pm i \frac{\sqrt{3}}{3^{1}}$

Evaluate the exponent.

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

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

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

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.

$x = \frac{i \sqrt{3}}{3}$

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

$x = - \frac{i \sqrt{3}}{3}$

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

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

Step 3

Substitute $\frac{i \sqrt{3}}{3}$ for $x$ .

$y = 3$

Step 4

List all of the solutions.

$y = 3, x = \frac{i \sqrt{3}}{3}$

$y = 3, x = - \frac{i \sqrt{3}}{3}$

Step 5

Solve the System of Equations y=-3x^2+2 and y=3

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