Algebra Examples

$x^{2} + y^{2} = 100$ , $y^{2} = x + 10$

Step 1

Solve for $y$ in $y^{2} = x + 10$ .

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Step 1.1

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

$y = \pm \sqrt{x + 10}$

$x^{2} + y^{2} = 100$

Step 1.2

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

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Step 1.2.1

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

$y = \sqrt{x + 10}$

$x^{2} + y^{2} = 100$

Step 1.2.2

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

$y = - \sqrt{x + 10}$

$x^{2} + y^{2} = 100$

Step 1.2.3

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

$y = \sqrt{x + 10}$

$y = - \sqrt{x + 10}$

$x^{2} + y^{2} = 100$

$y = \sqrt{x + 10}$

$y = - \sqrt{x + 10}$

$x^{2} + y^{2} = 100$

$y = \sqrt{x + 10}$

$y = - \sqrt{x + 10}$

$x^{2} + y^{2} = 100$

Step 2

Solve the system $y = \sqrt{x + 10}, x^{2} + y^{2} = 100$ .

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Step 2.1

Replace all occurrences of $y$ with $\sqrt{x + 10}$ in each equation.

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Step 2.1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 100$ with $\sqrt{x + 10}$ .

$x^{2} + {(\sqrt{x + 10})}^{2} = 100$

$y = \sqrt{x + 10}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify each term.

$x^{2} + x + 10 = 100$

$y = \sqrt{x + 10}$

$x^{2} + x + 10 = 100$

$y = \sqrt{x + 10}$

$x^{2} + x + 10 = 100$

$y = \sqrt{x + 10}$

Step 2.2

Solve for $x$ in $x^{2} + x + 10 = 100$ .

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Step 2.2.1

Subtract $100$ from both sides of the equation.

$x^{2} + x + 10 - 100 = 0$

$y = \sqrt{x + 10}$

Step 2.2.2

Subtract $100$ from $10$ .

$x^{2} + x - 90 = 0$

$y = \sqrt{x + 10}$

Step 2.2.3

Factor $x^{2} + x - 90$ using the AC method.

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Step 2.2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 90$ and whose sum is $1$ .

$- 9, 10$

$y = \sqrt{x + 10}$

Step 2.2.3.2

Write the factored form using these integers.

$(x - 9) (x + 10) = 0$

$y = \sqrt{x + 10}$

$(x - 9) (x + 10) = 0$

$y = \sqrt{x + 10}$

Step 2.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 9 = 0$

$x + 10 = 0$

$y = \sqrt{x + 10}$

Step 2.2.5

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

$y = \sqrt{x + 10}$

Step 2.2.5.2

Add $9$ to both sides of the equation.

$x = 9$

$y = \sqrt{x + 10}$

$x = 9$

$y = \sqrt{x + 10}$

Step 2.2.6

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 2.2.6.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

$y = \sqrt{x + 10}$

Step 2.2.6.2

Subtract $10$ from both sides of the equation.

$x = - 10$

$y = \sqrt{x + 10}$

$x = - 10$

$y = \sqrt{x + 10}$

Step 2.2.7

The final solution is all the values that make $(x - 9) (x + 10) = 0$ true.

$x = 9, - 10$

$y = \sqrt{x + 10}$

$x = 9, - 10$

$y = \sqrt{x + 10}$

Step 2.3

Replace all occurrences of $x$ with $9$ in each equation.

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Step 2.3.1

Replace all occurrences of $x$ in $y = \sqrt{x + 10}$ with $9$ .

$y = \sqrt{(9) + 10}$

$x = 9$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Add $9$ and $10$ .

$y = \sqrt{19}$

$x = 9$

$y = \sqrt{19}$

$x = 9$

$y = \sqrt{19}$

$x = 9$

Step 2.4

Replace all occurrences of $x$ with $- 10$ in each equation.

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Step 2.4.1

Replace all occurrences of $x$ in $y = \sqrt{x + 10}$ with $- 10$ .

$y = \sqrt{(- 10) + 10}$

$x = - 10$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{(- 10) + 10}$ .

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Step 2.4.2.1.1

Add $- 10$ and $10$ .

$y = \sqrt{0}$

$x = - 10$

Step 2.4.2.1.2

Rewrite $0$ as $0^{2}$ .

$y = \sqrt{0^{2}}$

$x = - 10$

Step 2.4.2.1.3

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

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

$y = 0$

$x = - 10$

Step 3

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

$(9, \sqrt{19})$

$(- 10, 0)$

$(9, - \sqrt{19})$

Step 4

The result can be shown in multiple forms.

Point Form:

$(9, \sqrt{19}), (- 10, 0), (9, - \sqrt{19})$

Equation Form:

$x = 9, y = \sqrt{19}$

$x = - 10, y = 0$

$x = 9, y = - \sqrt{19}$

Step 5