Algebra Examples

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

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} - 4 x + 2 = - \frac{3}{4} x - 1$

Step 2

Solve $x^{2} - 4 x + 2 = - \frac{3}{4} x - 1$ for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $x$ and $\frac{3}{4}$ .

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

Step 2.1.2

Move $3$ to the left of $x$ .

$x^{2} - 4 x + 2 = - \frac{3 x}{4} - 1$

Step 2.2

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

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Step 2.2.1

Add $\frac{3 x}{4}$ to both sides of the equation.

$x^{2} - 4 x + 2 + \frac{3 x}{4} = - 1$

Step 2.2.2

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.2.3

Combine $- 4 x$ and $\frac{4}{4}$ .

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Find the common denominator.

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Step 2.2.5.1

Write $x^{2}$ as a fraction with denominator $1$ .

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

Step 2.2.5.2

Multiply $\frac{x^{2}}{1}$ by $\frac{4}{4}$ .

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

Step 2.2.5.3

Multiply $\frac{x^{2}}{1}$ by $\frac{4}{4}$ .

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

Step 2.2.5.4

Write $2$ as a fraction with denominator $1$ .

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

Step 2.2.5.5

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

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

Step 2.2.5.6

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

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

Step 2.2.6

Combine the numerators over the common denominator.

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

Step 2.2.7

Simplify each term.

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Step 2.2.7.1

Move $4$ to the left of $x^{2}$ .

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

Step 2.2.7.2

Multiply $4$ by $- 4$ .

$\frac{4 x^{2} - 16 x + 3 x + 2 \cdot 4}{4} = - 1$

Step 2.2.7.3

Multiply $2$ by $4$ .

$\frac{4 x^{2} - 16 x + 3 x + 8}{4} = - 1$

Step 2.2.8

Add $- 16 x$ and $3 x$ .

$\frac{4 x^{2} - 13 x + 8}{4} = - 1$

Step 2.3

Multiply both sides by $4$ .

$\frac{4 x^{2} - 13 x + 8}{4} \cdot 4 = - 1 \cdot 4$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $4$ .

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Step 2.4.1.1.1

Cancel the common factor.

$\frac{4 x^{2} - 13 x + 8}{4} \cdot 4 = - 1 \cdot 4$

Step 2.4.1.1.2

Rewrite the expression.

$4 x^{2} - 13 x + 8 = - 1 \cdot 4$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $- 1$ by $4$ .

$4 x^{2} - 13 x + 8 = - 4$

Step 2.5

Solve for $x$ .

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Step 2.5.1

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

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Step 2.5.1.1

Add $4$ to both sides of the equation.

$4 x^{2} - 13 x + 8 + 4 = 0$

Step 2.5.1.2

Add $8$ and $4$ .

$4 x^{2} - 13 x + 12 = 0$

Step 2.5.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = - \frac{3}{4} x - 1$

Step 2.5.3

Substitute the values $a = 4$ , $b = - 13$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (4 \cdot 12)}}{2 \cdot 4} + y = - \frac{3}{4} x - 1$

Step 2.5.4

Simplify.

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Step 2.5.4.1

Simplify the numerator.

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Step 2.5.4.1.1

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 4 \cdot 12}}{2 \cdot 4}$

Step 2.5.4.1.2

Multiply $- 4 \cdot 4 \cdot 12$ .

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Step 2.5.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{13 \pm \sqrt{169 - 16 \cdot 12}}{2 \cdot 4}$

Step 2.5.4.1.2.2

Multiply $- 16$ by $12$ .

$x = \frac{13 \pm \sqrt{169 - 192}}{2 \cdot 4}$

Step 2.5.4.1.3

Subtract $192$ from $169$ .

$x = \frac{13 \pm \sqrt{- 23}}{2 \cdot 4}$

Step 2.5.4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{13 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 4}$

Step 2.5.4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{13 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 4}$

Step 2.5.4.1.6

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

$x = \frac{13 \pm i \sqrt{23}}{2 \cdot 4}$

Step 2.5.4.2

Multiply $2$ by $4$ .

$x = \frac{13 \pm i \sqrt{23}}{8}$

Step 2.5.5

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

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Step 2.5.5.1

Simplify the numerator.

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Step 2.5.5.1.1

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 4 \cdot 12}}{2 \cdot 4}$

Step 2.5.5.1.2

Multiply $- 4 \cdot 4 \cdot 12$ .

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Step 2.5.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{13 \pm \sqrt{169 - 16 \cdot 12}}{2 \cdot 4}$

Step 2.5.5.1.2.2

Multiply $- 16$ by $12$ .

$x = \frac{13 \pm \sqrt{169 - 192}}{2 \cdot 4}$

Step 2.5.5.1.3

Subtract $192$ from $169$ .

$x = \frac{13 \pm \sqrt{- 23}}{2 \cdot 4}$

Step 2.5.5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{13 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 4}$

Step 2.5.5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{13 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 4}$

Step 2.5.5.1.6

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

$x = \frac{13 \pm i \sqrt{23}}{2 \cdot 4}$

Step 2.5.5.2

Multiply $2$ by $4$ .

$x = \frac{13 \pm i \sqrt{23}}{8}$

Step 2.5.5.3

Change the $\pm$ to $+$ .

$x = \frac{13 + i \sqrt{23}}{8}$

Step 2.5.6

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

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Step 2.5.6.1

Simplify the numerator.

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Step 2.5.6.1.1

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 4 \cdot 12}}{2 \cdot 4}$

Step 2.5.6.1.2

Multiply $- 4 \cdot 4 \cdot 12$ .

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Step 2.5.6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{13 \pm \sqrt{169 - 16 \cdot 12}}{2 \cdot 4}$

Step 2.5.6.1.2.2

Multiply $- 16$ by $12$ .

$x = \frac{13 \pm \sqrt{169 - 192}}{2 \cdot 4}$

Step 2.5.6.1.3

Subtract $192$ from $169$ .

$x = \frac{13 \pm \sqrt{- 23}}{2 \cdot 4}$

Step 2.5.6.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{13 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 4}$

Step 2.5.6.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{13 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 4}$

Step 2.5.6.1.6

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

$x = \frac{13 \pm i \sqrt{23}}{2 \cdot 4}$

Step 2.5.6.2

Multiply $2$ by $4$ .

$x = \frac{13 \pm i \sqrt{23}}{8}$

Step 2.5.6.3

Change the $\pm$ to $-$ .

$x = \frac{13 - i \sqrt{23}}{8}$

Step 2.5.7

The final answer is the combination of both solutions.

$x = \frac{13 + i \sqrt{23}}{8}, \frac{13 - i \sqrt{23}}{8}$

Step 3

Evaluate $y$ when $x = \frac{13 + i \sqrt{23}}{8}$ .

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Step 3.1

Substitute $\frac{13 + i \sqrt{23}}{8}$ for $x$ .

$y = - \frac{3}{4} \cdot \frac{13 + i \sqrt{23}}{8} - 1$

Step 3.2

Substitute $\frac{13 + i \sqrt{23}}{8}$ for $x$ in $y = - \frac{3}{4} \cdot \frac{13 + i \sqrt{23}}{8} - 1$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = - \frac{3}{4} \cdot \frac{13 + i \sqrt{23}}{8} - 1$

Step 3.2.2

Simplify $- \frac{3}{4} \cdot \frac{13 + i \sqrt{23}}{8} - 1$ .

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $- \frac{3}{4} \cdot \frac{13 + i \sqrt{23}}{8}$ .

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Step 3.2.2.1.1.1

Multiply $\frac{13 + i \sqrt{23}}{8}$ by $\frac{3}{4}$ .

$y = - \frac{(13 + i \sqrt{23}) \cdot 3}{8 \cdot 4} - 1$

Step 3.2.2.1.1.2

Multiply $8$ by $4$ .

$y = - \frac{(13 + i \sqrt{23}) \cdot 3}{32} - 1$

Step 3.2.2.1.2

Move $3$ to the left of $13 + i \sqrt{23}$ .

$y = - \frac{3 (13 + i \sqrt{23})}{32} - 1$

Step 3.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$y = - \frac{3 (13 + i \sqrt{23})}{32} - 1 \cdot \frac{32}{32}$

Step 3.2.2.3

Combine $- 1$ and $\frac{32}{32}$ .

$y = - \frac{3 (13 + i \sqrt{23})}{32} + \frac{- 1 \cdot 32}{32}$

Step 3.2.2.4

Combine the numerators over the common denominator.

$y = \frac{- 71 - 3 i \sqrt{23}}{32}$

Step 3.2.2.5

Rewrite $- 71$ as $- 1 (71)$ .

$y = \frac{- 1 (71) - 3 i \sqrt{23}}{32}$

Step 3.2.2.6

Factor $- 1$ out of $- 3 i \sqrt{23}$ .

$y = \frac{- 1 (71) - (3 i \sqrt{23})}{32}$

Step 3.2.2.7

Factor $- 1$ out of $- 1 (71) - (3 i \sqrt{23})$ .

$y = \frac{- 1 (71 + 3 i \sqrt{23})}{32}$

Step 3.2.2.8

Move the negative in front of the fraction.

$y = - \frac{71 + 3 i \sqrt{23}}{32}$

Step 4

Evaluate $y$ when $x = \frac{13 - i \sqrt{23}}{8}$ .

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Step 4.1

Substitute $\frac{13 - i \sqrt{23}}{8}$ for $x$ .

$y = - \frac{3}{4} \cdot \frac{13 - i \sqrt{23}}{8} - 1$

Step 4.2

Substitute $\frac{13 - i \sqrt{23}}{8}$ for $x$ in $y = - \frac{3}{4} \cdot \frac{13 - i \sqrt{23}}{8} - 1$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = - \frac{3}{4} \cdot \frac{13 - i \sqrt{23}}{8} - 1$

Step 4.2.2

Simplify $- \frac{3}{4} \cdot \frac{13 - i \sqrt{23}}{8} - 1$ .

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $- \frac{3}{4} \cdot \frac{13 - i \sqrt{23}}{8}$ .

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Step 4.2.2.1.1.1

Multiply $\frac{13 - i \sqrt{23}}{8}$ by $\frac{3}{4}$ .

$y = - \frac{(13 - i \sqrt{23}) \cdot 3}{8 \cdot 4} - 1$

Step 4.2.2.1.1.2

Multiply $8$ by $4$ .

$y = - \frac{(13 - i \sqrt{23}) \cdot 3}{32} - 1$

Step 4.2.2.1.2

Move $3$ to the left of $13 - i \sqrt{23}$ .

$y = - \frac{3 (13 - i \sqrt{23})}{32} - 1$

Step 4.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$y = - \frac{3 (13 - i \sqrt{23})}{32} - 1 \cdot \frac{32}{32}$

Step 4.2.2.3

Combine $- 1$ and $\frac{32}{32}$ .

$y = - \frac{3 (13 - i \sqrt{23})}{32} + \frac{- 1 \cdot 32}{32}$

Step 4.2.2.4

Combine the numerators over the common denominator.

$y = \frac{- 71 + 3 i \sqrt{23}}{32}$

Step 4.2.2.5

Rewrite $- 71$ as $- 1 (71)$ .

$y = \frac{- 1 (71) + 3 i \sqrt{23}}{32}$

Step 4.2.2.6

Factor $- 1$ out of $3 i \sqrt{23}$ .

$y = \frac{- 1 (71) - (- 3 i \sqrt{23})}{32}$

Step 4.2.2.7

Factor $- 1$ out of $- 1 (71) - (- 3 i \sqrt{23})$ .

$y = \frac{- 1 (71 - 3 i \sqrt{23})}{32}$

Step 4.2.2.8

Move the negative in front of the fraction.

$y = - \frac{71 - 3 i \sqrt{23}}{32}$

Step 5

List all of the solutions.

$y = - \frac{71 + 3 i \sqrt{23}}{32}, x = \frac{13 + i \sqrt{23}}{8}$

$y = - \frac{71 - 3 i \sqrt{23}}{32}, x = \frac{13 - i \sqrt{23}}{8}$

Step 6