Algebra Examples

$y = - 3 x^{2} + 2$ and $y = 4 x^{2} + 3$

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

Solve $- 3 x^{2} + 2 = 4 x^{2} + 3$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Subtract $4 x^{2}$ from both sides of the equation.

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

Step 2.1.2

Subtract $4 x^{2}$ from $- 3 x^{2}$ .

$- 7 x^{2} + 2 = 3$

Step 2.2

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

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Step 2.2.1

Subtract $2$ from both sides of the equation.

$- 7 x^{2} = 3 - 2$

Step 2.2.2

Subtract $2$ from $3$ .

$- 7 x^{2} = 1$

Step 2.3

Divide each term in $- 7 x^{2} = 1$ by $- 7$ and simplify.

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Step 2.3.1

Divide each term in $- 7 x^{2} = 1$ by $- 7$ .

$\frac{- 7 x^{2}}{- 7} = \frac{1}{- 7}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 7$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 7 x^{2}}{- 7} = \frac{1}{- 7}$

Step 2.3.2.1.2

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

$x^{2} = \frac{1}{- 7}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{7}$

Step 2.4

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

$x = \pm \sqrt{- \frac{1}{7}}$

Step 2.5

Simplify $\pm \sqrt{- \frac{1}{7}}$ .

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Step 2.5.1

Rewrite $- \frac{1}{7}$ as $i^{2} \frac{1^{2}}{7}$ .

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Step 2.5.1.1

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

$x = \pm \sqrt{i^{2} \frac{1}{7}}$

Step 2.5.1.2

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

$x = \pm \sqrt{i^{2} \frac{1^{2}}{7}}$

Step 2.5.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{7}}$

Step 2.5.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{7}}$

Step 2.5.4

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{7}}$

Step 2.5.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{7}}$

Step 2.5.6

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm i (\frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 2.5.7

Combine and simplify the denominator.

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Step 2.5.7.1

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm i \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.5.7.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 2.5.7.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 2.5.7.4

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

$x = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.5.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.5.7.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.5.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.5.7.6.2

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

$x = \pm i \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.5.7.6.3

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

$x = \pm i \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.5.7.6.4

Cancel the common factor of $2$ .

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Step 2.5.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.5.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{7}}{7^{1}}$

Step 2.5.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{7}}{7}$

Step 2.5.8

Combine $i$ and $\frac{\sqrt{7}}{7}$ .

$x = \pm \frac{i \sqrt{7}}{7}$

Step 2.6

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

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Step 2.6.1

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

$x = \frac{i \sqrt{7}}{7}$

Step 2.6.2

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

$x = - \frac{i \sqrt{7}}{7}$

Step 2.6.3

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

$x = \frac{i \sqrt{7}}{7}, - \frac{i \sqrt{7}}{7}$

Step 3

Evaluate $y$ when $x = \frac{i \sqrt{7}}{7}$ .

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Step 3.1

Substitute $\frac{i \sqrt{7}}{7}$ for $x$ .

$y = 4 {(\frac{i \sqrt{7}}{7})}^{2} + 3$

Step 3.2

Simplify $4 {(\frac{i \sqrt{7}}{7})}^{2} + 3$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.1.1

Apply the product rule to $\frac{i \sqrt{7}}{7}$ .

$y = 4 \frac{{(i \sqrt{7})}^{2}}{7^{2}} + 3$

Step 3.2.1.1.2

Apply the product rule to $i \sqrt{7}$ .

$y = 4 \frac{i^{2} {\sqrt{7}}^{2}}{7^{2}} + 3$

Step 3.2.1.2

Simplify the numerator.

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Step 3.2.1.2.1

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

$y = 4 \frac{- 1 {\sqrt{7}}^{2}}{7^{2}} + 3$

Step 3.2.1.2.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.2.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = 4 \frac{- 1 {(7^{\frac{1}{2}})}^{2}}{7^{2}} + 3$

Step 3.2.1.2.2.2

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

$y = 4 \frac{- 1 \cdot 7^{\frac{1}{2} \cdot 2}}{7^{2}} + 3$

Step 3.2.1.2.2.3

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

$y = 4 \frac{- 1 \cdot 7^{\frac{2}{2}}}{7^{2}} + 3$

Step 3.2.1.2.2.4

Cancel the common factor of $2$ .

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Step 3.2.1.2.2.4.1

Cancel the common factor.

$y = 4 \frac{- 1 \cdot 7^{\frac{2}{2}}}{7^{2}} + 3$

Step 3.2.1.2.2.4.2

Rewrite the expression.

$y = 4 \frac{- 1 \cdot 7^{1}}{7^{2}} + 3$

Step 3.2.1.2.2.5

Evaluate the exponent.

$y = 4 \frac{- 1 \cdot 7}{7^{2}} + 3$

Step 3.2.1.3

Raise $7$ to the power of $2$ .

$y = 4 \frac{- 1 \cdot 7}{49} + 3$

Step 3.2.1.4

Multiply $- 1$ by $7$ .

$y = 4 (\frac{- 7}{49}) + 3$

Step 3.2.1.5

Cancel the common factor of $- 7$ and $49$ .

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Step 3.2.1.5.1

Factor $7$ out of $- 7$ .

$y = 4 \frac{7 (- 1)}{49} + 3$

Step 3.2.1.5.2

Cancel the common factors.

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Step 3.2.1.5.2.1

Factor $7$ out of $49$ .

$y = 4 \frac{7 \cdot - 1}{7 \cdot 7} + 3$

Step 3.2.1.5.2.2

Cancel the common factor.

$y = 4 \frac{7 \cdot - 1}{7 \cdot 7} + 3$

Step 3.2.1.5.2.3

Rewrite the expression.

$y = 4 (\frac{- 1}{7}) + 3$

Step 3.2.1.6

Move the negative in front of the fraction.

$y = 4 (- \frac{1}{7}) + 3$

Step 3.2.1.7

Multiply $4 (- \frac{1}{7})$ .

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Step 3.2.1.7.1

Multiply $- 1$ by $4$ .

$y = - 4 (\frac{1}{7}) + 3$

Step 3.2.1.7.2

Combine $- 4$ and $\frac{1}{7}$ .

$y = \frac{- 4}{7} + 3$

Step 3.2.1.8

Move the negative in front of the fraction.

$y = - \frac{4}{7} + 3$

Step 3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{4}{7} + 3 \cdot \frac{7}{7}$

Step 3.2.3

Combine $3$ and $\frac{7}{7}$ .

$y = - \frac{4}{7} + \frac{3 \cdot 7}{7}$

Step 3.2.4

Combine the numerators over the common denominator.

$y = \frac{- 4 + 3 \cdot 7}{7}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $3$ by $7$ .

$y = \frac{- 4 + 21}{7}$

Step 3.2.5.2

Add $- 4$ and $21$ .

$y = \frac{17}{7}$

Step 4

List all of the solutions.

$y = \frac{17}{7}, x = \frac{i \sqrt{7}}{7}$

$y = \frac{17}{7}, x = - \frac{i \sqrt{7}}{7}$

Step 5