Algebra Examples

$y + 1 = - 2 x^{2} + 4$ $2 x - y = 5$

Step 1

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

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Step 1.1

Subtract $1$ from both sides of the equation.

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

$2 x - y = 5$

Step 1.2

Subtract $1$ from $4$ .

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

$2 x - y = 5$

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

$2 x - y = 5$

Step 2

Replace all occurrences of $y$ with $- 2 x^{2} + 3$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x - y = 5$ with $- 2 x^{2} + 3$ .

$2 x - (- 2 x^{2} + 3) = 5$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$2 x - (- 2 x^{2}) - 1 \cdot 3 = 5$

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

Step 2.2.1.2

Multiply $- 2$ by $- 1$ .

$2 x + 2 x^{2} - 1 \cdot 3 = 5$

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

Step 2.2.1.3

Multiply $- 1$ by $3$ .

$2 x + 2 x^{2} - 3 = 5$

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

$2 x + 2 x^{2} - 3 = 5$

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

$2 x + 2 x^{2} - 3 = 5$

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

$2 x + 2 x^{2} - 3 = 5$

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

Step 3

Solve for $x$ in $2 x + 2 x^{2} - 3 = 5$ .

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Step 3.1

Subtract $5$ from both sides of the equation.

$2 x + 2 x^{2} - 3 - 5 = 0$

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

Step 3.2

Subtract $5$ from $- 3$ .

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

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

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $2$ out of $2 x + 2 x^{2} - 8$ .

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Step 3.3.1.1

Factor $2$ out of $2 x$ .

$2 (x) + 2 x^{2} - 8 = 0$

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

Step 3.3.1.2

Factor $2$ out of $2 x^{2}$ .

$2 (x) + 2 (x^{2}) - 8 = 0$

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

Step 3.3.1.3

Factor $2$ out of $- 8$ .

$2 (x) + 2 x^{2} + 2 \cdot - 4 = 0$

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

Step 3.3.1.4

Factor $2$ out of $2 (x) + 2 x^{2}$ .

$2 (x + x^{2}) + 2 \cdot - 4 = 0$

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

Step 3.3.1.5

Factor $2$ out of $2 (x + x^{2}) + 2 \cdot - 4$ .

$2 (x + x^{2} - 4) = 0$

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

$2 (x + x^{2} - 4) = 0$

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

Step 3.3.2

Reorder terms.

$2 (x^{2} + x - 4) = 0$

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

$2 (x^{2} + x - 4) = 0$

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

Step 3.4

Divide each term in $2 (x^{2} + x - 4) = 0$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 (x^{2} + x - 4) = 0$ by $2$ .

$\frac{2 (x^{2} + x - 4)}{2} = \frac{0}{2}$

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + x - 4)}{2} = \frac{0}{2}$

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

Step 3.4.2.1.2

Divide $x^{2} + x - 4$ by $1$ .

$x^{2} + x - 4 = \frac{0}{2}$

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

$x^{2} + x - 4 = \frac{0}{2}$

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

$x^{2} + x - 4 = \frac{0}{2}$

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $2$ .

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

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

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

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

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

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

Step 3.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

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

Step 3.6

Substitute the values $a = 1$ , $b = 1$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

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

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

One to any power is one.

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

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

Step 3.7.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $1$ .

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

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

Step 3.7.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

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

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

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

Step 3.7.1.3

Add $1$ and $16$ .

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

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

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

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

Step 3.7.2

Multiply $2$ by $1$ .

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

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

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

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

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

One to any power is one.

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

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

Step 3.8.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $1$ .

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

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

Step 3.8.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

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

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

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

Step 3.8.1.3

Add $1$ and $16$ .

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

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

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

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

Step 3.8.2

Multiply $2$ by $1$ .

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

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

Step 3.8.3

Change the $\pm$ to $+$ .

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

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

Step 3.8.4

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

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

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

Step 3.8.5

Factor $- 1$ out of $\sqrt{17}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{17})}{2}$

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

Step 3.8.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{17})$ .

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

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

Step 3.8.7

Move the negative in front of the fraction.

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

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

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

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

Step 3.9

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

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Step 3.9.1

Simplify the numerator.

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Step 3.9.1.1

One to any power is one.

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

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

Step 3.9.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.9.1.2.1

Multiply $- 4$ by $1$ .

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

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

Step 3.9.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

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

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

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

Step 3.9.1.3

Add $1$ and $16$ .

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

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

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

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

Step 3.9.2

Multiply $2$ by $1$ .

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

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

Step 3.9.3

Change the $\pm$ to $-$ .

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

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

Step 3.9.4

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

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

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

Step 3.9.5

Factor $- 1$ out of $- \sqrt{17}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{17})}{2}$

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

Step 3.9.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{17})$ .

$x = \frac{- 1 (1 + \sqrt{17})}{2}$

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

Step 3.9.7

Move the negative in front of the fraction.

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

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

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

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

Step 3.10

The final answer is the combination of both solutions.

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

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

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

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

Step 4

Replace all occurrences of $x$ with $- \frac{1 - \sqrt{17}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 2 x^{2} + 3$ with $- \frac{1 - \sqrt{17}}{2}$ .

$y = - 2 {(- \frac{1 - \sqrt{17}}{2})}^{2} + 3$

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 2 {(- \frac{1 - \sqrt{17}}{2})}^{2} + 3$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

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Step 4.2.1.1.1.1

Apply the product rule to $- \frac{1 - \sqrt{17}}{2}$ .

$y = - 2 ({(- 1)}^{2} {(\frac{1 - \sqrt{17}}{2})}^{2}) + 3$

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

Step 4.2.1.1.1.2

Apply the product rule to $\frac{1 - \sqrt{17}}{2}$ .

$y = - 2 ({(- 1)}^{2} (\frac{{(1 - \sqrt{17})}^{2}}{2^{2}})) + 3$

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

$y = - 2 ({(- 1)}^{2} (\frac{{(1 - \sqrt{17})}^{2}}{2^{2}})) + 3$

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

Step 4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = - 2 (1 (\frac{{(1 - \sqrt{17})}^{2}}{2^{2}})) + 3$

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

Step 4.2.1.1.3

Multiply $\frac{{(1 - \sqrt{17})}^{2}}{2^{2}}$ by $1$ .

$y = - 2 \frac{{(1 - \sqrt{17})}^{2}}{2^{2}} + 3$

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

Step 4.2.1.1.4

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

$y = - 2 \frac{{(1 - \sqrt{17})}^{2}}{4} + 3$

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

Step 4.2.1.1.5

Cancel the common factor of $2$ .

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Step 4.2.1.1.5.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) (\frac{{(1 - \sqrt{17})}^{2}}{4}) + 3$

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

Step 4.2.1.1.5.2

Factor $2$ out of $4$ .

$y = 2 \cdot (- 1 \frac{{(1 - \sqrt{17})}^{2}}{2 \cdot 2}) + 3$

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

Step 4.2.1.1.5.3

Cancel the common factor.

$y = 2 \cdot (- 1 \frac{{(1 - \sqrt{17})}^{2}}{2 \cdot 2}) + 3$

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

Step 4.2.1.1.5.4

Rewrite the expression.

$y = - 1 \frac{{(1 - \sqrt{17})}^{2}}{2} + 3$

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

$y = - 1 \frac{{(1 - \sqrt{17})}^{2}}{2} + 3$

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

Step 4.2.1.1.6

Rewrite ${(1 - \sqrt{17})}^{2}$ as $(1 - \sqrt{17}) (1 - \sqrt{17})$ .

$y = - 1 \frac{(1 - \sqrt{17}) (1 - \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.7

Expand $(1 - \sqrt{17}) (1 - \sqrt{17})$ using the FOIL Method.

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Step 4.2.1.1.7.1

Apply the distributive property.

$y = - 1 \frac{1 (1 - \sqrt{17}) - \sqrt{17} (1 - \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.7.2

Apply the distributive property.

$y = - 1 \frac{1 \cdot 1 + 1 (- \sqrt{17}) - \sqrt{17} (1 - \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.7.3

Apply the distributive property.

$y = - 1 \frac{1 \cdot 1 + 1 (- \sqrt{17}) - \sqrt{17} \cdot 1 - \sqrt{17} (- \sqrt{17})}{2} + 3$

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

$y = - 1 \frac{1 \cdot 1 + 1 (- \sqrt{17}) - \sqrt{17} \cdot 1 - \sqrt{17} (- \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.8

Simplify and combine like terms.

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Step 4.2.1.1.8.1

Simplify each term.

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Step 4.2.1.1.8.1.1

Multiply $1$ by $1$ .

$y = - 1 \frac{1 + 1 (- \sqrt{17}) - \sqrt{17} \cdot 1 - \sqrt{17} (- \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.8.1.2

Multiply $- \sqrt{17}$ by $1$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} \cdot 1 - \sqrt{17} (- \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.8.1.3

Multiply $- 1$ by $1$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} - \sqrt{17} (- \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.8.1.4

Multiply $- \sqrt{17} (- \sqrt{17})$ .

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Step 4.2.1.1.8.1.4.1

Multiply $- 1$ by $- 1$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 1 \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 4.2.1.1.8.1.4.2

Multiply $\sqrt{17}$ by $1$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 4.2.1.1.8.1.4.3

Raise $\sqrt{17}$ to the power of $1$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 4.2.1.1.8.1.4.4

Raise $\sqrt{17}$ to the power of $1$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 4.2.1.1.8.1.4.5

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

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + {\sqrt{17}}^{1 + 1}}{2} + 3$

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

Step 4.2.1.1.8.1.4.6

Add $1$ and $1$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + {\sqrt{17}}^{2}}{2} + 3$

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

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + {\sqrt{17}}^{2}}{2} + 3$

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

Step 4.2.1.1.8.1.5

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 4.2.1.1.8.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + {(17^{\frac{1}{2}})}^{2}}{2} + 3$

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

Step 4.2.1.1.8.1.5.2

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

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17^{\frac{1}{2} \cdot 2}}{2} + 3$

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

Step 4.2.1.1.8.1.5.3

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

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17^{\frac{2}{2}}}{2} + 3$

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

Step 4.2.1.1.8.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.8.1.5.4.1

Cancel the common factor.

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17^{\frac{2}{2}}}{2} + 3$

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

Step 4.2.1.1.8.1.5.4.2

Rewrite the expression.

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17}{2} + 3$

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

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17}{2} + 3$

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

Step 4.2.1.1.8.1.5.5

Evaluate the exponent.

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17}{2} + 3$

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

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17}{2} + 3$

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

$y = - 1 \frac{1 - \sqrt{17} - \sqrt{17} + 17}{2} + 3$

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

Step 4.2.1.1.8.2

Add $1$ and $17$ .

$y = - 1 \frac{18 - \sqrt{17} - \sqrt{17}}{2} + 3$

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

Step 4.2.1.1.8.3

Subtract $\sqrt{17}$ from $- \sqrt{17}$ .

$y = - 1 \frac{18 - 2 \sqrt{17}}{2} + 3$

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

$y = - 1 \frac{18 - 2 \sqrt{17}}{2} + 3$

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

Step 4.2.1.1.9

Cancel the common factor of $18 - 2 \sqrt{17}$ and $2$ .

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Step 4.2.1.1.9.1

Factor $2$ out of $18$ .

$y = - 1 \frac{2 \cdot 9 - 2 \sqrt{17}}{2} + 3$

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

Step 4.2.1.1.9.2

Factor $2$ out of $- 2 \sqrt{17}$ .

$y = - 1 \frac{2 \cdot 9 + 2 (- \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.9.3

Factor $2$ out of $2 (9) + 2 (- \sqrt{17})$ .

$y = - 1 \frac{2 (9 - \sqrt{17})}{2} + 3$

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

Step 4.2.1.1.9.4

Cancel the common factors.

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Step 4.2.1.1.9.4.1

Factor $2$ out of $2$ .

$y = - 1 \frac{2 (9 - \sqrt{17})}{2 (1)} + 3$

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

Step 4.2.1.1.9.4.2

Cancel the common factor.

$y = - 1 \frac{2 (9 - \sqrt{17})}{2 \cdot 1} + 3$

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

Step 4.2.1.1.9.4.3

Rewrite the expression.

$y = - 1 \frac{9 - \sqrt{17}}{1} + 3$

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

Step 4.2.1.1.9.4.4

Divide $9 - \sqrt{17}$ by $1$ .

$y = - 1 (9 - \sqrt{17}) + 3$

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

$y = - 1 (9 - \sqrt{17}) + 3$

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

$y = - 1 (9 - \sqrt{17}) + 3$

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

Step 4.2.1.1.10

Apply the distributive property.

$y = - 1 \cdot 9 - 1 (- \sqrt{17}) + 3$

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

Step 4.2.1.1.11

Multiply $- 1$ by $9$ .

$y = - 9 - 1 (- \sqrt{17}) + 3$

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

Step 4.2.1.1.12

Multiply $- 1 (- \sqrt{17})$ .

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Step 4.2.1.1.12.1

Multiply $- 1$ by $- 1$ .

$y = - 9 + 1 \sqrt{17} + 3$

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

Step 4.2.1.1.12.2

Multiply $\sqrt{17}$ by $1$ .

$y = - 9 + \sqrt{17} + 3$

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

$y = - 9 + \sqrt{17} + 3$

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

$y = - 9 + \sqrt{17} + 3$

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

Step 4.2.1.2

Add $- 9$ and $3$ .

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

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

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

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

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

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

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

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

Step 5

Replace all occurrences of $x$ with $- \frac{1 + \sqrt{17}}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 2 x^{2} + 3$ with $- \frac{1 + \sqrt{17}}{2}$ .

$y = - 2 {(- \frac{1 + \sqrt{17}}{2})}^{2} + 3$

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 2 {(- \frac{1 + \sqrt{17}}{2})}^{2} + 3$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{1 + \sqrt{17}}{2}$ .

$y = - 2 ({(- 1)}^{2} {(\frac{1 + \sqrt{17}}{2})}^{2}) + 3$

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

Step 5.2.1.1.1.2

Apply the product rule to $\frac{1 + \sqrt{17}}{2}$ .

$y = - 2 ({(- 1)}^{2} (\frac{{(1 + \sqrt{17})}^{2}}{2^{2}})) + 3$

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

$y = - 2 ({(- 1)}^{2} (\frac{{(1 + \sqrt{17})}^{2}}{2^{2}})) + 3$

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

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = - 2 (1 (\frac{{(1 + \sqrt{17})}^{2}}{2^{2}})) + 3$

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

Step 5.2.1.1.3

Multiply $\frac{{(1 + \sqrt{17})}^{2}}{2^{2}}$ by $1$ .

$y = - 2 \frac{{(1 + \sqrt{17})}^{2}}{2^{2}} + 3$

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

Step 5.2.1.1.4

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

$y = - 2 \frac{{(1 + \sqrt{17})}^{2}}{4} + 3$

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

Step 5.2.1.1.5

Cancel the common factor of $2$ .

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Step 5.2.1.1.5.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) (\frac{{(1 + \sqrt{17})}^{2}}{4}) + 3$

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

Step 5.2.1.1.5.2

Factor $2$ out of $4$ .

$y = 2 \cdot (- 1 \frac{{(1 + \sqrt{17})}^{2}}{2 \cdot 2}) + 3$

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

Step 5.2.1.1.5.3

Cancel the common factor.

$y = 2 \cdot (- 1 \frac{{(1 + \sqrt{17})}^{2}}{2 \cdot 2}) + 3$

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

Step 5.2.1.1.5.4

Rewrite the expression.

$y = - 1 \frac{{(1 + \sqrt{17})}^{2}}{2} + 3$

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

$y = - 1 \frac{{(1 + \sqrt{17})}^{2}}{2} + 3$

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

Step 5.2.1.1.6

Rewrite ${(1 + \sqrt{17})}^{2}$ as $(1 + \sqrt{17}) (1 + \sqrt{17})$ .

$y = - 1 \frac{(1 + \sqrt{17}) (1 + \sqrt{17})}{2} + 3$

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

Step 5.2.1.1.7

Expand $(1 + \sqrt{17}) (1 + \sqrt{17})$ using the FOIL Method.

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Step 5.2.1.1.7.1

Apply the distributive property.

$y = - 1 \frac{1 (1 + \sqrt{17}) + \sqrt{17} (1 + \sqrt{17})}{2} + 3$

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

Step 5.2.1.1.7.2

Apply the distributive property.

$y = - 1 \frac{1 \cdot 1 + 1 \sqrt{17} + \sqrt{17} (1 + \sqrt{17})}{2} + 3$

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

Step 5.2.1.1.7.3

Apply the distributive property.

$y = - 1 \frac{1 \cdot 1 + 1 \sqrt{17} + \sqrt{17} \cdot 1 + \sqrt{17} \sqrt{17}}{2} + 3$

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

$y = - 1 \frac{1 \cdot 1 + 1 \sqrt{17} + \sqrt{17} \cdot 1 + \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 5.2.1.1.8

Simplify and combine like terms.

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Step 5.2.1.1.8.1

Simplify each term.

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Step 5.2.1.1.8.1.1

Multiply $1$ by $1$ .

$y = - 1 \frac{1 + 1 \sqrt{17} + \sqrt{17} \cdot 1 + \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 5.2.1.1.8.1.2

Multiply $\sqrt{17}$ by $1$ .

$y = - 1 \frac{1 + \sqrt{17} + \sqrt{17} \cdot 1 + \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 5.2.1.1.8.1.3

Multiply $\sqrt{17}$ by $1$ .

$y = - 1 \frac{1 + \sqrt{17} + \sqrt{17} + \sqrt{17} \sqrt{17}}{2} + 3$

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

Step 5.2.1.1.8.1.4

Combine using the product rule for radicals.

$y = - 1 \frac{1 + \sqrt{17} + \sqrt{17} + \sqrt{17 \cdot 17}}{2} + 3$

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

Step 5.2.1.1.8.1.5

Multiply $17$ by $17$ .

$y = - 1 \frac{1 + \sqrt{17} + \sqrt{17} + \sqrt{289}}{2} + 3$

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

Step 5.2.1.1.8.1.6

Rewrite $289$ as $17^{2}$ .

$y = - 1 \frac{1 + \sqrt{17} + \sqrt{17} + \sqrt{17^{2}}}{2} + 3$

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

Step 5.2.1.1.8.1.7

Pull terms out from under the radical, assuming positive real numbers.

$y = - 1 \frac{1 + \sqrt{17} + \sqrt{17} + 17}{2} + 3$

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

$y = - 1 \frac{1 + \sqrt{17} + \sqrt{17} + 17}{2} + 3$

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

Step 5.2.1.1.8.2

Add $1$ and $17$ .

$y = - 1 \frac{18 + \sqrt{17} + \sqrt{17}}{2} + 3$

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

Step 5.2.1.1.8.3

Add $\sqrt{17}$ and $\sqrt{17}$ .

$y = - 1 \frac{18 + 2 \sqrt{17}}{2} + 3$

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

$y = - 1 \frac{18 + 2 \sqrt{17}}{2} + 3$

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

Step 5.2.1.1.9

Cancel the common factor of $18 + 2 \sqrt{17}$ and $2$ .

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Step 5.2.1.1.9.1

Factor $2$ out of $18$ .

$y = - 1 \frac{2 \cdot 9 + 2 \sqrt{17}}{2} + 3$

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

Step 5.2.1.1.9.2

Factor $2$ out of $2 \sqrt{17}$ .

$y = - 1 \frac{2 \cdot 9 + 2 (\sqrt{17})}{2} + 3$

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

Step 5.2.1.1.9.3

Factor $2$ out of $2 (9) + 2 (\sqrt{17})$ .

$y = - 1 \frac{2 (9 + \sqrt{17})}{2} + 3$

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

Step 5.2.1.1.9.4

Cancel the common factors.

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Step 5.2.1.1.9.4.1

Factor $2$ out of $2$ .

$y = - 1 \frac{2 (9 + \sqrt{17})}{2 (1)} + 3$

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

Step 5.2.1.1.9.4.2

Cancel the common factor.

$y = - 1 \frac{2 (9 + \sqrt{17})}{2 \cdot 1} + 3$

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

Step 5.2.1.1.9.4.3

Rewrite the expression.

$y = - 1 \frac{9 + \sqrt{17}}{1} + 3$

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

Step 5.2.1.1.9.4.4

Divide $9 + \sqrt{17}$ by $1$ .

$y = - 1 (9 + \sqrt{17}) + 3$

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

$y = - 1 (9 + \sqrt{17}) + 3$

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

$y = - 1 (9 + \sqrt{17}) + 3$

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

Step 5.2.1.1.10

Apply the distributive property.

$y = - 1 \cdot 9 - 1 \sqrt{17} + 3$

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

Step 5.2.1.1.11

Multiply $- 1$ by $9$ .

$y = - 9 - 1 \sqrt{17} + 3$

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

Step 5.2.1.1.12

Rewrite $- 1 \sqrt{17}$ as $- \sqrt{17}$ .

$y = - 9 - \sqrt{17} + 3$

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

$y = - 9 - \sqrt{17} + 3$

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

Step 5.2.1.2

Add $- 9$ and $3$ .

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

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

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

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

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

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

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

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

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- \frac{1 - \sqrt{17}}{2}, - 6 + \sqrt{17})$

$(- \frac{1 + \sqrt{17}}{2}, - 6 - \sqrt{17})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- \frac{1 - \sqrt{17}}{2}, - 6 + \sqrt{17}), (- \frac{1 + \sqrt{17}}{2}, - 6 - \sqrt{17})$

Equation Form:

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

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

Step 8