Algebra Examples

$y = x^{2} - 4 x - 30$ , $2 y + 8 x = - 10$

Step 1

Replace all occurrences of $y$ with $x^{2} - 4 x - 30$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $2 y + 8 x = - 10$ with $x^{2} - 4 x - 30$ .

$2 (x^{2} - 4 x - 30) + 8 x = - 10$

$y = x^{2} - 4 x - 30$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $2 (x^{2} - 4 x - 30) + 8 x$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the distributive property.

$2 x^{2} + 2 (- 4 x) + 2 \cdot - 30 + 8 x = - 10$

$y = x^{2} - 4 x - 30$

Step 1.2.1.1.2

Simplify.

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Step 1.2.1.1.2.1

Multiply $- 4$ by $2$ .

$2 x^{2} - 8 x + 2 \cdot - 30 + 8 x = - 10$

$y = x^{2} - 4 x - 30$

Step 1.2.1.1.2.2

Multiply $2$ by $- 30$ .

$2 x^{2} - 8 x - 60 + 8 x = - 10$

$y = x^{2} - 4 x - 30$

$2 x^{2} - 8 x - 60 + 8 x = - 10$

$y = x^{2} - 4 x - 30$

$2 x^{2} - 8 x - 60 + 8 x = - 10$

$y = x^{2} - 4 x - 30$

Step 1.2.1.2

Combine the opposite terms in $2 x^{2} - 8 x - 60 + 8 x$ .

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Step 1.2.1.2.1

Add $- 8 x$ and $8 x$ .

$2 x^{2} + 0 - 60 = - 10$

$y = x^{2} - 4 x - 30$

Step 1.2.1.2.2

Add $2 x^{2}$ and $0$ .

$2 x^{2} - 60 = - 10$

$y = x^{2} - 4 x - 30$

$2 x^{2} - 60 = - 10$

$y = x^{2} - 4 x - 30$

$2 x^{2} - 60 = - 10$

$y = x^{2} - 4 x - 30$

$2 x^{2} - 60 = - 10$

$y = x^{2} - 4 x - 30$

$2 x^{2} - 60 = - 10$

$y = x^{2} - 4 x - 30$

Step 2

Solve for $x$ in $2 x^{2} - 60 = - 10$ .

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Step 2.1

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

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Step 2.1.1

Add $60$ to both sides of the equation.

$2 x^{2} = - 10 + 60$

$y = x^{2} - 4 x - 30$

Step 2.1.2

Add $- 10$ and $60$ .

$2 x^{2} = 50$

$y = x^{2} - 4 x - 30$

$2 x^{2} = 50$

$y = x^{2} - 4 x - 30$

Step 2.2

Divide each term in $2 x^{2} = 50$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x^{2} = 50$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{50}{2}$

$y = x^{2} - 4 x - 30$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{50}{2}$

$y = x^{2} - 4 x - 30$

Step 2.2.2.1.2

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

$x^{2} = \frac{50}{2}$

$y = x^{2} - 4 x - 30$

$x^{2} = \frac{50}{2}$

$y = x^{2} - 4 x - 30$

$x^{2} = \frac{50}{2}$

$y = x^{2} - 4 x - 30$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $50$ by $2$ .

$x^{2} = 25$

$y = x^{2} - 4 x - 30$

$x^{2} = 25$

$y = x^{2} - 4 x - 30$

$x^{2} = 25$

$y = x^{2} - 4 x - 30$

Step 2.3

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

$x = \pm \sqrt{25}$

$y = x^{2} - 4 x - 30$

Step 2.4

Simplify $\pm \sqrt{25}$ .

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Step 2.4.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

$y = x^{2} - 4 x - 30$

Step 2.4.2

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

$x = \pm 5$

$y = x^{2} - 4 x - 30$

$x = \pm 5$

$y = x^{2} - 4 x - 30$

Step 2.5

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

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Step 2.5.1

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

$x = 5$

$y = x^{2} - 4 x - 30$

Step 2.5.2

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

$x = - 5$

$y = x^{2} - 4 x - 30$

Step 2.5.3

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

$x = 5, - 5$

$y = x^{2} - 4 x - 30$

$x = 5, - 5$

$y = x^{2} - 4 x - 30$

$x = 5, - 5$

$y = x^{2} - 4 x - 30$

Step 3

Replace all occurrences of $x$ with $5$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2} - 4 x - 30$ with $5$ .

$y = {(5)}^{2} - 4 \cdot 5 - 30$

$x = 5$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(5)}^{2} - 4 \cdot 5 - 30$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Raise $5$ to the power of $2$ .

$y = 25 - 4 \cdot 5 - 30$

$x = 5$

Step 3.2.1.1.2

Multiply $- 4$ by $5$ .

$y = 25 - 20 - 30$

$x = 5$

$y = 25 - 20 - 30$

$x = 5$

Step 3.2.1.2

Simplify by subtracting numbers.

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Step 3.2.1.2.1

Subtract $20$ from $25$ .

$y = 5 - 30$

$x = 5$

Step 3.2.1.2.2

Subtract $30$ from $5$ .

$y = - 25$

$x = 5$

$y = - 25$

$x = 5$

$y = - 25$

$x = 5$

$y = - 25$

$x = 5$

$y = - 25$

$x = 5$

Step 4

Replace all occurrences of $x$ with $- 5$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2} - 4 x - 30$ with $- 5$ .

$y = {(- 5)}^{2} - 4 \cdot - 5 - 30$

$x = - 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(- 5)}^{2} - 4 \cdot - 5 - 30$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Raise $- 5$ to the power of $2$ .

$y = 25 - 4 \cdot - 5 - 30$

$x = - 5$

Step 4.2.1.1.2

Multiply $- 4$ by $- 5$ .

$y = 25 + 20 - 30$

$x = - 5$

$y = 25 + 20 - 30$

$x = - 5$

Step 4.2.1.2

Simplify by adding and subtracting.

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Step 4.2.1.2.1

Add $25$ and $20$ .

$y = 45 - 30$

$x = - 5$

Step 4.2.1.2.2

Subtract $30$ from $45$ .

$y = 15$

$x = - 5$

$y = 15$

$x = - 5$

$y = 15$

$x = - 5$

$y = 15$

$x = - 5$

$y = 15$

$x = - 5$

Step 5

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

$(5, - 25)$

$(- 5, 15)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(5, - 25), (- 5, 15)$

Equation Form:

$x = 5, y = - 25$

$x = - 5, y = 15$

Step 7