Algebra Examples

$y = 2 x^{2} + 3$ $y = 2 x - 6$

Step 1

Eliminate the equal sides of each equation and combine.

$2 x^{2} + 3 = 2 x - 6$

Step 2

Solve $2 x^{2} + 3 = 2 x - 6$ for $x$ .

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Step 2.1

Subtract $2 x$ from both sides of the equation.

$2 x^{2} + 3 - 2 x = - 6$

Step 2.2

Move all terms to the left side of the equation and simplify.

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Step 2.2.1

Add $6$ to both sides of the equation.

$2 x^{2} + 3 - 2 x + 6 = 0$

Step 2.2.2

Add $3$ and $6$ .

$2 x^{2} - 2 x + 9 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 2 x - 6$

Step 2.4

Substitute the values $a = 2$ , $b = - 2$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot 9)}}{2 \cdot 2} + y = 2 x - 6$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$

Step 2.5.1.2

Multiply $- 4 \cdot 2 \cdot 9$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 9}}{2 \cdot 2}$

Step 2.5.1.2.2

Multiply $- 8$ by $9$ .

$x = \frac{2 \pm \sqrt{4 - 72}}{2 \cdot 2}$

Step 2.5.1.3

Subtract $72$ from $4$ .

$x = \frac{2 \pm \sqrt{- 68}}{2 \cdot 2}$

Step 2.5.1.4

Rewrite $- 68$ as $- 1 (68)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 68}}{2 \cdot 2}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (68)}$ as $\sqrt{- 1} \cdot \sqrt{68}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{68}}{2 \cdot 2}$

Step 2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{68}}{2 \cdot 2}$

Step 2.5.1.7

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 2.5.1.7.1

Factor $4$ out of $68$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (17)}}{2 \cdot 2}$

Step 2.5.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 17}}{2 \cdot 2}$

Step 2.5.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{17})}{2 \cdot 2}$

Step 2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{17}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i \sqrt{17}}{4}$

Step 2.5.3

Simplify $\frac{2 \pm 2 i \sqrt{17}}{4}$ .

$x = \frac{1 \pm i \sqrt{17}}{2}$

Step 2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$

Step 2.6.1.2

Multiply $- 4 \cdot 2 \cdot 9$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 9}}{2 \cdot 2}$

Step 2.6.1.2.2

Multiply $- 8$ by $9$ .

$x = \frac{2 \pm \sqrt{4 - 72}}{2 \cdot 2}$

Step 2.6.1.3

Subtract $72$ from $4$ .

$x = \frac{2 \pm \sqrt{- 68}}{2 \cdot 2}$

Step 2.6.1.4

Rewrite $- 68$ as $- 1 (68)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 68}}{2 \cdot 2}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (68)}$ as $\sqrt{- 1} \cdot \sqrt{68}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{68}}{2 \cdot 2}$

Step 2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{68}}{2 \cdot 2}$

Step 2.6.1.7

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 2.6.1.7.1

Factor $4$ out of $68$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (17)}}{2 \cdot 2}$

Step 2.6.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 17}}{2 \cdot 2}$

Step 2.6.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{17})}{2 \cdot 2}$

Step 2.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{17}}{2 \cdot 2}$

Step 2.6.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i \sqrt{17}}{4}$

Step 2.6.3

Simplify $\frac{2 \pm 2 i \sqrt{17}}{4}$ .

$x = \frac{1 \pm i \sqrt{17}}{2}$

Step 2.6.4

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{17}}{2}$

Step 2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$

Step 2.7.1.2

Multiply $- 4 \cdot 2 \cdot 9$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 9}}{2 \cdot 2}$

Step 2.7.1.2.2

Multiply $- 8$ by $9$ .

$x = \frac{2 \pm \sqrt{4 - 72}}{2 \cdot 2}$

Step 2.7.1.3

Subtract $72$ from $4$ .

$x = \frac{2 \pm \sqrt{- 68}}{2 \cdot 2}$

Step 2.7.1.4

Rewrite $- 68$ as $- 1 (68)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 68}}{2 \cdot 2}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (68)}$ as $\sqrt{- 1} \cdot \sqrt{68}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{68}}{2 \cdot 2}$

Step 2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{68}}{2 \cdot 2}$

Step 2.7.1.7

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 2.7.1.7.1

Factor $4$ out of $68$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (17)}}{2 \cdot 2}$

Step 2.7.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 17}}{2 \cdot 2}$

Step 2.7.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{17})}{2 \cdot 2}$

Step 2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{17}}{2 \cdot 2}$

Step 2.7.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i \sqrt{17}}{4}$

Step 2.7.3

Simplify $\frac{2 \pm 2 i \sqrt{17}}{4}$ .

$x = \frac{1 \pm i \sqrt{17}}{2}$

Step 2.7.4

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{17}}{2}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{17}}{2}, \frac{1 - i \sqrt{17}}{2}$

Step 3

Evaluate $y$ when $x = \frac{1 + i \sqrt{17}}{2}$ .

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Step 3.1

Substitute $\frac{1 + i \sqrt{17}}{2}$ for $x$ .

$y = 2 (\frac{1 + i \sqrt{17}}{2}) - 6$

Step 3.2

Substitute $\frac{1 + i \sqrt{17}}{2}$ for $x$ in $y = 2 (\frac{1 + i \sqrt{17}}{2}) - 6$ and solve for $y$ .

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Step 3.2.1

Multiply $2$ by $\frac{1 + i \sqrt{17}}{2}$ .

$y = 2 (\frac{1 + i \sqrt{17}}{2}) - 6$

Step 3.2.2

Simplify $2 (\frac{1 + i \sqrt{17}}{2}) - 6$ .

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$y = 2 \frac{1 + i \sqrt{17}}{2} - 6$

Step 3.2.2.1.2

Rewrite the expression.

$y = 1 + i \sqrt{17} - 6$

Step 3.2.2.2

Simplify the expression.

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Step 3.2.2.2.1

Subtract $6$ from $1$ .

$y = i \sqrt{17} - 5$

Step 3.2.2.2.2

Reorder $i \sqrt{17}$ and $- 5$ .

$y = - 5 + i \sqrt{17}$

Step 4

Evaluate $y$ when $x = \frac{1 - i \sqrt{17}}{2}$ .

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Step 4.1

Substitute $\frac{1 - i \sqrt{17}}{2}$ for $x$ .

$y = 2 (\frac{1 - i \sqrt{17}}{2}) - 6$

Step 4.2

Substitute $\frac{1 - i \sqrt{17}}{2}$ for $x$ in $y = 2 (\frac{1 - i \sqrt{17}}{2}) - 6$ and solve for $y$ .

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Step 4.2.1

Multiply $2$ by $\frac{1 - i \sqrt{17}}{2}$ .

$y = 2 (\frac{1 - i \sqrt{17}}{2}) - 6$

Step 4.2.2

Simplify $2 (\frac{1 - i \sqrt{17}}{2}) - 6$ .

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$y = 2 \frac{1 - i \sqrt{17}}{2} - 6$

Step 4.2.2.1.2

Rewrite the expression.

$y = 1 - i \sqrt{17} - 6$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Subtract $6$ from $1$ .

$y = - i \sqrt{17} - 5$

Step 4.2.2.2.2

Reorder $- i \sqrt{17}$ and $- 5$ .

$y = - 5 - i \sqrt{17}$

Step 5

List all of the solutions.

$y = - 5 + i \sqrt{17}, x = \frac{1 + i \sqrt{17}}{2}$

$y = - 5 - i \sqrt{17}, x = \frac{1 - i \sqrt{17}}{2}$

Step 6