Algebra Examples

$y = - 7 x^{2} - 9 x + 6$ $y = \frac{1}{2} x + 11$

Step 1

Eliminate the equal sides of each equation and combine.

$- 7 x^{2} - 9 x + 6 = \frac{1}{2} x + 11$

Step 2

Solve $- 7 x^{2} - 9 x + 6 = \frac{1}{2} x + 11$ for $x$ .

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Step 2.1

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

$- 7 x^{2} - 9 x + 6 = \frac{x}{2} + 11$

Step 2.2

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

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Step 2.2.1

Subtract $\frac{x}{2}$ from both sides of the equation.

$- 7 x^{2} - 9 x + 6 - \frac{x}{2} = 11$

Step 2.2.2

To write $- 9 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 7 x^{2} - 9 x \cdot \frac{2}{2} - \frac{x}{2} + 6 = 11$

Step 2.2.3

Combine $- 9 x$ and $\frac{2}{2}$ .

$- 7 x^{2} + \frac{- 9 x \cdot 2}{2} - \frac{x}{2} + 6 = 11$

Step 2.2.4

Combine the numerators over the common denominator.

$- 7 x^{2} + \frac{- 9 x \cdot 2 - x}{2} + 6 = 11$

Step 2.2.5

Find the common denominator.

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Step 2.2.5.1

Write $- 7 x^{2}$ as a fraction with denominator $1$ .

$\frac{- 7 x^{2}}{1} + \frac{- 9 x \cdot 2 - x}{2} + 6 = 11$

Step 2.2.5.2

Multiply $\frac{- 7 x^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{- 7 x^{2}}{1} \cdot \frac{2}{2} + \frac{- 9 x \cdot 2 - x}{2} + 6 = 11$

Step 2.2.5.3

Multiply $\frac{- 7 x^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{- 7 x^{2} \cdot 2}{2} + \frac{- 9 x \cdot 2 - x}{2} + 6 = 11$

Step 2.2.5.4

Write $6$ as a fraction with denominator $1$ .

$\frac{- 7 x^{2} \cdot 2}{2} + \frac{- 9 x \cdot 2 - x}{2} + \frac{6}{1} = 11$

Step 2.2.5.5

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$\frac{- 7 x^{2} \cdot 2}{2} + \frac{- 9 x \cdot 2 - x}{2} + \frac{6}{1} \cdot \frac{2}{2} = 11$

Step 2.2.5.6

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$\frac{- 7 x^{2} \cdot 2}{2} + \frac{- 9 x \cdot 2 - x}{2} + \frac{6 \cdot 2}{2} = 11$

Step 2.2.6

Combine the numerators over the common denominator.

$\frac{- 7 x^{2} \cdot 2 - 9 x \cdot 2 - x + 6 \cdot 2}{2} = 11$

Step 2.2.7

Simplify each term.

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Step 2.2.7.1

Multiply $2$ by $- 7$ .

$\frac{- 14 x^{2} - 9 x \cdot 2 - x + 6 \cdot 2}{2} = 11$

Step 2.2.7.2

Multiply $2$ by $- 9$ .

$\frac{- 14 x^{2} - 18 x - x + 6 \cdot 2}{2} = 11$

Step 2.2.7.3

Multiply $6$ by $2$ .

$\frac{- 14 x^{2} - 18 x - x + 12}{2} = 11$

Step 2.2.8

Subtract $x$ from $- 18 x$ .

$\frac{- 14 x^{2} - 19 x + 12}{2} = 11$

Step 2.2.9

Factor $- 1$ out of $- 14 x^{2}$ .

$\frac{- (14 x^{2}) - 19 x + 12}{2} = 11$

Step 2.2.10

Factor $- 1$ out of $- 19 x$ .

$\frac{- (14 x^{2}) - (19 x) + 12}{2} = 11$

Step 2.2.11

Factor $- 1$ out of $- (14 x^{2}) - (19 x)$ .

$\frac{- (14 x^{2} + 19 x) + 12}{2} = 11$

Step 2.2.12

Rewrite $12$ as $- 1 (- 12)$ .

$\frac{- (14 x^{2} + 19 x) - 1 (- 12)}{2} = 11$

Step 2.2.13

Factor $- 1$ out of $- (14 x^{2} + 19 x) - 1 (- 12)$ .

$\frac{- (14 x^{2} + 19 x - 12)}{2} = 11$

Step 2.2.14

Rewrite $- (14 x^{2} + 19 x - 12)$ as $- 1 (14 x^{2} + 19 x - 12)$ .

$\frac{- 1 (14 x^{2} + 19 x - 12)}{2} = 11$

Step 2.2.15

Move the negative in front of the fraction.

$- \frac{14 x^{2} + 19 x - 12}{2} = 11$

Step 2.3

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{14 x^{2} + 19 x - 12}{2}) = - 2 \cdot 11$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify $- 2 (- \frac{14 x^{2} + 19 x - 12}{2})$ .

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Step 2.4.1.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1.1

Move the leading negative in $- \frac{14 x^{2} + 19 x - 12}{2}$ into the numerator.

$- 2 \frac{- (14 x^{2} + 19 x - 12)}{2} = - 2 \cdot 11$

Step 2.4.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- (14 x^{2} + 19 x - 12)}{2} = - 2 \cdot 11$

Step 2.4.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- (14 x^{2} + 19 x - 12)}{2} = - 2 \cdot 11$

Step 2.4.1.1.1.4

Rewrite the expression.

$- - (14 x^{2} + 19 x - 12) = - 2 \cdot 11$

Step 2.4.1.1.2

Multiply.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (14 x^{2} + 19 x - 12) = - 2 \cdot 11$

Step 2.4.1.1.2.2

Multiply $14 x^{2} + 19 x - 12$ by $1$ .

$14 x^{2} + 19 x - 12 = - 2 \cdot 11$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $- 2$ by $11$ .

$14 x^{2} + 19 x - 12 = - 22$

Step 2.5

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

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Step 2.5.1

Add $22$ to both sides of the equation.

$14 x^{2} + 19 x - 12 + 22 = 0$

Step 2.5.2

Add $- 12$ and $22$ .

$14 x^{2} + 19 x + 10 = 0$

Step 2.6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = \frac{1}{2} x + 11$

Step 2.7

Substitute the values $a = 14$ , $b = 19$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{- 19 \pm \sqrt{19^{2} - 4 \cdot (14 \cdot 10)}}{2 \cdot 14} + y = \frac{1}{2} x + 11$

Step 2.8

Simplify.

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Raise $19$ to the power of $2$ .

$x = \frac{- 19 \pm \sqrt{361 - 4 \cdot 14 \cdot 10}}{2 \cdot 14}$

Step 2.8.1.2

Multiply $- 4 \cdot 14 \cdot 10$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{- 19 \pm \sqrt{361 - 56 \cdot 10}}{2 \cdot 14}$

Step 2.8.1.2.2

Multiply $- 56$ by $10$ .

$x = \frac{- 19 \pm \sqrt{361 - 560}}{2 \cdot 14}$

Step 2.8.1.3

Subtract $560$ from $361$ .

$x = \frac{- 19 \pm \sqrt{- 199}}{2 \cdot 14}$

Step 2.8.1.4

Rewrite $- 199$ as $- 1 (199)$ .

$x = \frac{- 19 \pm \sqrt{- 1 \cdot 199}}{2 \cdot 14}$

Step 2.8.1.5

Rewrite $\sqrt{- 1 (199)}$ as $\sqrt{- 1} \cdot \sqrt{199}$ .

$x = \frac{- 19 \pm \sqrt{- 1} \cdot \sqrt{199}}{2 \cdot 14}$

Step 2.8.1.6

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

$x = \frac{- 19 \pm i \sqrt{199}}{2 \cdot 14}$

Step 2.8.2

Multiply $2$ by $14$ .

$x = \frac{- 19 \pm i \sqrt{199}}{28}$

Step 2.9

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

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Step 2.9.1

Simplify the numerator.

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Step 2.9.1.1

Raise $19$ to the power of $2$ .

$x = \frac{- 19 \pm \sqrt{361 - 4 \cdot 14 \cdot 10}}{2 \cdot 14}$

Step 2.9.1.2

Multiply $- 4 \cdot 14 \cdot 10$ .

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Step 2.9.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{- 19 \pm \sqrt{361 - 56 \cdot 10}}{2 \cdot 14}$

Step 2.9.1.2.2

Multiply $- 56$ by $10$ .

$x = \frac{- 19 \pm \sqrt{361 - 560}}{2 \cdot 14}$

Step 2.9.1.3

Subtract $560$ from $361$ .

$x = \frac{- 19 \pm \sqrt{- 199}}{2 \cdot 14}$

Step 2.9.1.4

Rewrite $- 199$ as $- 1 (199)$ .

$x = \frac{- 19 \pm \sqrt{- 1 \cdot 199}}{2 \cdot 14}$

Step 2.9.1.5

Rewrite $\sqrt{- 1 (199)}$ as $\sqrt{- 1} \cdot \sqrt{199}$ .

$x = \frac{- 19 \pm \sqrt{- 1} \cdot \sqrt{199}}{2 \cdot 14}$

Step 2.9.1.6

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

$x = \frac{- 19 \pm i \sqrt{199}}{2 \cdot 14}$

Step 2.9.2

Multiply $2$ by $14$ .

$x = \frac{- 19 \pm i \sqrt{199}}{28}$

Step 2.9.3

Change the $\pm$ to $+$ .

$x = \frac{- 19 + i \sqrt{199}}{28}$

Step 2.9.4

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{- 1 \cdot 19 + i \sqrt{199}}{28}$

Step 2.9.5

Factor $- 1$ out of $i \sqrt{199}$ .

$x = \frac{- 1 \cdot 19 - (- i \sqrt{199})}{28}$

Step 2.9.6

Factor $- 1$ out of $- 1 (19) - (- i \sqrt{199})$ .

$x = \frac{- 1 (19 - i \sqrt{199})}{28}$

Step 2.9.7

Move the negative in front of the fraction.

$x = - \frac{19 - i \sqrt{199}}{28}$

Step 2.10

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

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Step 2.10.1

Simplify the numerator.

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Step 2.10.1.1

Raise $19$ to the power of $2$ .

$x = \frac{- 19 \pm \sqrt{361 - 4 \cdot 14 \cdot 10}}{2 \cdot 14}$

Step 2.10.1.2

Multiply $- 4 \cdot 14 \cdot 10$ .

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Step 2.10.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{- 19 \pm \sqrt{361 - 56 \cdot 10}}{2 \cdot 14}$

Step 2.10.1.2.2

Multiply $- 56$ by $10$ .

$x = \frac{- 19 \pm \sqrt{361 - 560}}{2 \cdot 14}$

Step 2.10.1.3

Subtract $560$ from $361$ .

$x = \frac{- 19 \pm \sqrt{- 199}}{2 \cdot 14}$

Step 2.10.1.4

Rewrite $- 199$ as $- 1 (199)$ .

$x = \frac{- 19 \pm \sqrt{- 1 \cdot 199}}{2 \cdot 14}$

Step 2.10.1.5

Rewrite $\sqrt{- 1 (199)}$ as $\sqrt{- 1} \cdot \sqrt{199}$ .

$x = \frac{- 19 \pm \sqrt{- 1} \cdot \sqrt{199}}{2 \cdot 14}$

Step 2.10.1.6

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

$x = \frac{- 19 \pm i \sqrt{199}}{2 \cdot 14}$

Step 2.10.2

Multiply $2$ by $14$ .

$x = \frac{- 19 \pm i \sqrt{199}}{28}$

Step 2.10.3

Change the $\pm$ to $-$ .

$x = \frac{- 19 - i \sqrt{199}}{28}$

Step 2.10.4

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{- 1 \cdot 19 - i \sqrt{199}}{28}$

Step 2.10.5

Factor $- 1$ out of $- i \sqrt{199}$ .

$x = \frac{- 1 \cdot 19 - (i \sqrt{199})}{28}$

Step 2.10.6

Factor $- 1$ out of $- 1 (19) - (i \sqrt{199})$ .

$x = \frac{- 1 (19 + i \sqrt{199})}{28}$

Step 2.10.7

Move the negative in front of the fraction.

$x = - \frac{19 + i \sqrt{199}}{28}$

Step 2.11

The final answer is the combination of both solutions.

$x = - \frac{19 - i \sqrt{199}}{28}, - \frac{19 + i \sqrt{199}}{28}$

Step 3

Evaluate $y$ when $x = - \frac{19 - i \sqrt{199}}{28}$ .

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Step 3.1

Substitute $- \frac{19 - i \sqrt{199}}{28}$ for $x$ .

$y = \frac{1}{2} \cdot (- \frac{19 - i \sqrt{199}}{28}) + 11$

Step 3.2

Substitute $- \frac{19 - i \sqrt{199}}{28}$ for $x$ in $y = \frac{1}{2} \cdot (- \frac{19 - i \sqrt{199}}{28}) + 11$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = \frac{1}{2} \cdot (- 1 \frac{19 - i \sqrt{199}}{28}) + 11$

Step 3.2.2

Simplify $\frac{1}{2} \cdot (- 1 \frac{19 - i \sqrt{199}}{28}) + 11$ .

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Rewrite $- 1 \frac{19 - i \sqrt{199}}{28}$ as $- \frac{19 - i \sqrt{199}}{28}$ .

$y = \frac{1}{2} \cdot (- \frac{19 - i \sqrt{199}}{28}) + 11$

Step 3.2.2.1.2

Multiply $\frac{1}{2} (- \frac{19 - i \sqrt{199}}{28})$ .

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Step 3.2.2.1.2.1

Multiply $\frac{1}{2}$ by $\frac{19 - i \sqrt{199}}{28}$ .

$y = - \frac{19 - i \sqrt{199}}{2 \cdot 28} + 11$

Step 3.2.2.1.2.2

Multiply $2$ by $28$ .

$y = - \frac{19 - i \sqrt{199}}{56} + 11$

Step 3.2.2.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{56}{56}$ .

$y = - \frac{19 - i \sqrt{199}}{56} + 11 \cdot \frac{56}{56}$

Step 3.2.2.3

Combine $11$ and $\frac{56}{56}$ .

$y = - \frac{19 - i \sqrt{199}}{56} + \frac{11 \cdot 56}{56}$

Step 3.2.2.4

Combine the numerators over the common denominator.

$y = \frac{597 + \sqrt{199} i}{56}$

Step 4

Evaluate $y$ when $x = - \frac{19 + i \sqrt{199}}{28}$ .

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Step 4.1

Substitute $- \frac{19 + i \sqrt{199}}{28}$ for $x$ .

$y = \frac{1}{2} \cdot (- \frac{19 + i \sqrt{199}}{28}) + 11$

Step 4.2

Substitute $- \frac{19 + i \sqrt{199}}{28}$ for $x$ in $y = \frac{1}{2} \cdot (- \frac{19 + i \sqrt{199}}{28}) + 11$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = \frac{1}{2} \cdot (- 1 \frac{19 + i \sqrt{199}}{28}) + 11$

Step 4.2.2

Simplify $\frac{1}{2} \cdot (- 1 \frac{19 + i \sqrt{199}}{28}) + 11$ .

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Rewrite $- 1 \frac{19 + i \sqrt{199}}{28}$ as $- \frac{19 + i \sqrt{199}}{28}$ .

$y = \frac{1}{2} \cdot (- \frac{19 + i \sqrt{199}}{28}) + 11$

Step 4.2.2.1.2

Multiply $\frac{1}{2} (- \frac{19 + i \sqrt{199}}{28})$ .

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Step 4.2.2.1.2.1

Multiply $\frac{1}{2}$ by $\frac{19 + i \sqrt{199}}{28}$ .

$y = - \frac{19 + i \sqrt{199}}{2 \cdot 28} + 11$

Step 4.2.2.1.2.2

Multiply $2$ by $28$ .

$y = - \frac{19 + i \sqrt{199}}{56} + 11$

Step 4.2.2.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{56}{56}$ .

$y = - \frac{19 + i \sqrt{199}}{56} + 11 \cdot \frac{56}{56}$

Step 4.2.2.3

Combine $11$ and $\frac{56}{56}$ .

$y = - \frac{19 + i \sqrt{199}}{56} + \frac{11 \cdot 56}{56}$

Step 4.2.2.4

Combine the numerators over the common denominator.

$y = \frac{597 - i \sqrt{199}}{56}$

Step 5

List all of the solutions.

$y = \frac{597 + \sqrt{199} i}{56}, x = - \frac{19 - i \sqrt{199}}{28}$

$y = \frac{597 - i \sqrt{199}}{56}, x = - \frac{19 + i \sqrt{199}}{28}$

Step 6