Algebra Examples

$y = - 2 x^{2} + 3 x + 82$ $16 = 3 x - y$

Step 1

Replace all occurrences of $y$ with $- 2 x^{2} + 3 x + 82$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $16 = 3 x - y$ with $- 2 x^{2} + 3 x + 82$ .

$16 = 3 x - (- 2 x^{2} + 3 x + 82)$

$y = - 2 x^{2} + 3 x + 82$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $3 x - (- 2 x^{2} + 3 x + 82)$ .

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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.

$16 = 3 x - (- 2 x^{2}) - (3 x) - 1 \cdot 82$

$y = - 2 x^{2} + 3 x + 82$

Step 1.2.1.1.2

Simplify.

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Step 1.2.1.1.2.1

Multiply $- 2$ by $- 1$ .

$16 = 3 x + 2 x^{2} - (3 x) - 1 \cdot 82$

$y = - 2 x^{2} + 3 x + 82$

Step 1.2.1.1.2.2

Multiply $3$ by $- 1$ .

$16 = 3 x + 2 x^{2} - 3 x - 1 \cdot 82$

$y = - 2 x^{2} + 3 x + 82$

Step 1.2.1.1.2.3

Multiply $- 1$ by $82$ .

$16 = 3 x + 2 x^{2} - 3 x - 82$

$y = - 2 x^{2} + 3 x + 82$

$16 = 3 x + 2 x^{2} - 3 x - 82$

$y = - 2 x^{2} + 3 x + 82$

$16 = 3 x + 2 x^{2} - 3 x - 82$

$y = - 2 x^{2} + 3 x + 82$

Step 1.2.1.2

Combine the opposite terms in $3 x + 2 x^{2} - 3 x - 82$ .

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Step 1.2.1.2.1

Subtract $3 x$ from $3 x$ .

$16 = 2 x^{2} + 0 - 82$

$y = - 2 x^{2} + 3 x + 82$

Step 1.2.1.2.2

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

$16 = 2 x^{2} - 82$

$y = - 2 x^{2} + 3 x + 82$

$16 = 2 x^{2} - 82$

$y = - 2 x^{2} + 3 x + 82$

$16 = 2 x^{2} - 82$

$y = - 2 x^{2} + 3 x + 82$

$16 = 2 x^{2} - 82$

$y = - 2 x^{2} + 3 x + 82$

$16 = 2 x^{2} - 82$

$y = - 2 x^{2} + 3 x + 82$

Step 2

Solve for $x$ in $16 = 2 x^{2} - 82$ .

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Step 2.1

Rewrite the equation as $2 x^{2} - 82 = 16$ .

$2 x^{2} - 82 = 16$

$y = - 2 x^{2} + 3 x + 82$

Step 2.2

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

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Step 2.2.1

Add $82$ to both sides of the equation.

$2 x^{2} = 16 + 82$

$y = - 2 x^{2} + 3 x + 82$

Step 2.2.2

Add $16$ and $82$ .

$2 x^{2} = 98$

$y = - 2 x^{2} + 3 x + 82$

$2 x^{2} = 98$

$y = - 2 x^{2} + 3 x + 82$

Step 2.3

Divide each term in $2 x^{2} = 98$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 x^{2} = 98$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{98}{2}$

$y = - 2 x^{2} + 3 x + 82$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{98}{2}$

$y = - 2 x^{2} + 3 x + 82$

Step 2.3.2.1.2

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

$x^{2} = \frac{98}{2}$

$y = - 2 x^{2} + 3 x + 82$

$x^{2} = \frac{98}{2}$

$y = - 2 x^{2} + 3 x + 82$

$x^{2} = \frac{98}{2}$

$y = - 2 x^{2} + 3 x + 82$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $98$ by $2$ .

$x^{2} = 49$

$y = - 2 x^{2} + 3 x + 82$

$x^{2} = 49$

$y = - 2 x^{2} + 3 x + 82$

$x^{2} = 49$

$y = - 2 x^{2} + 3 x + 82$

Step 2.4

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

$x = \pm \sqrt{49}$

$y = - 2 x^{2} + 3 x + 82$

Step 2.5

Simplify $\pm \sqrt{49}$ .

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Step 2.5.1

Rewrite $49$ as $7^{2}$ .

$x = \pm \sqrt{7^{2}}$

$y = - 2 x^{2} + 3 x + 82$

Step 2.5.2

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

$x = \pm 7$

$y = - 2 x^{2} + 3 x + 82$

$x = \pm 7$

$y = - 2 x^{2} + 3 x + 82$

Step 2.6

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

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Step 2.6.1

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

$x = 7$

$y = - 2 x^{2} + 3 x + 82$

Step 2.6.2

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

$x = - 7$

$y = - 2 x^{2} + 3 x + 82$

Step 2.6.3

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

$x = 7, - 7$

$y = - 2 x^{2} + 3 x + 82$

$x = 7, - 7$

$y = - 2 x^{2} + 3 x + 82$

$x = 7, - 7$

$y = - 2 x^{2} + 3 x + 82$

Step 3

Replace all occurrences of $x$ with $7$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = - 2 x^{2} + 3 x + 82$ with $7$ .

$y = - 2 {(7)}^{2} + 3 (7) + 82$

$x = 7$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- 2 {(7)}^{2} + 3 (7) + 82$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Raise $7$ to the power of $2$ .

$y = - 2 \cdot 49 + 3 (7) + 82$

$x = 7$

Step 3.2.1.1.2

Multiply $- 2$ by $49$ .

$y = - 98 + 3 (7) + 82$

$x = 7$

Step 3.2.1.1.3

Multiply $3$ by $7$ .

$y = - 98 + 21 + 82$

$x = 7$

$y = - 98 + 21 + 82$

$x = 7$

Step 3.2.1.2

Simplify by adding numbers.

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Step 3.2.1.2.1

Add $- 98$ and $21$ .

$y = - 77 + 82$

$x = 7$

Step 3.2.1.2.2

Add $- 77$ and $82$ .

$y = 5$

$x = 7$

$y = 5$

$x = 7$

$y = 5$

$x = 7$

$y = 5$

$x = 7$

$y = 5$

$x = 7$

Step 4

Replace all occurrences of $x$ with $- 7$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 2 x^{2} + 3 x + 82$ with $- 7$ .

$y = - 2 {(- 7)}^{2} + 3 (- 7) + 82$

$x = - 7$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 2 {(- 7)}^{2} + 3 (- 7) + 82$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Raise $- 7$ to the power of $2$ .

$y = - 2 \cdot 49 + 3 (- 7) + 82$

$x = - 7$

Step 4.2.1.1.2

Multiply $- 2$ by $49$ .

$y = - 98 + 3 (- 7) + 82$

$x = - 7$

Step 4.2.1.1.3

Multiply $3$ by $- 7$ .

$y = - 98 - 21 + 82$

$x = - 7$

$y = - 98 - 21 + 82$

$x = - 7$

Step 4.2.1.2

Simplify by adding and subtracting.

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Step 4.2.1.2.1

Subtract $21$ from $- 98$ .

$y = - 119 + 82$

$x = - 7$

Step 4.2.1.2.2

Add $- 119$ and $82$ .

$y = - 37$

$x = - 7$

$y = - 37$

$x = - 7$

$y = - 37$

$x = - 7$

$y = - 37$

$x = - 7$

$y = - 37$

$x = - 7$

Step 5

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

$(7, 5)$

$(- 7, - 37)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(7, 5), (- 7, - 37)$

Equation Form:

$x = 7, y = 5$

$x = - 7, y = - 37$

Step 7