Algebra Examples

$y + 2 x + 1 = 0$ $4 y - 4 x^{2} - 12 x = - 7$

Step 1

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

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Step 1.1

Subtract $2 x$ from both sides of the equation.

$y + 1 = - 2 x$

$4 y - 4 x^{2} - 12 x = - 7$

Step 1.2

Subtract $1$ from both sides of the equation.

$y = - 2 x - 1$

$4 y - 4 x^{2} - 12 x = - 7$

$y = - 2 x - 1$

$4 y - 4 x^{2} - 12 x = - 7$

Step 2

Replace all occurrences of $y$ with $- 2 x - 1$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $4 y - 4 x^{2} - 12 x = - 7$ with $- 2 x - 1$ .

$4 (- 2 x - 1) - 4 x^{2} - 12 x = - 7$

$y = - 2 x - 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 (- 2 x - 1) - 4 x^{2} - 12 x$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$4 (- 2 x) + 4 \cdot - 1 - 4 x^{2} - 12 x = - 7$

$y = - 2 x - 1$

Step 2.2.1.1.2

Multiply $- 2$ by $4$ .

$- 8 x + 4 \cdot - 1 - 4 x^{2} - 12 x = - 7$

$y = - 2 x - 1$

Step 2.2.1.1.3

Multiply $4$ by $- 1$ .

$- 8 x - 4 - 4 x^{2} - 12 x = - 7$

$y = - 2 x - 1$

$- 8 x - 4 - 4 x^{2} - 12 x = - 7$

$y = - 2 x - 1$

Step 2.2.1.2

Subtract $12 x$ from $- 8 x$ .

$- 20 x - 4 - 4 x^{2} = - 7$

$y = - 2 x - 1$

$- 20 x - 4 - 4 x^{2} = - 7$

$y = - 2 x - 1$

$- 20 x - 4 - 4 x^{2} = - 7$

$y = - 2 x - 1$

$- 20 x - 4 - 4 x^{2} = - 7$

$y = - 2 x - 1$

Step 3

Solve for $x$ in $- 20 x - 4 - 4 x^{2} = - 7$ .

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Step 3.1

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

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Step 3.1.1

Add $7$ to both sides of the equation.

$- 20 x - 4 - 4 x^{2} + 7 = 0$

$y = - 2 x - 1$

Step 3.1.2

Add $- 4$ and $7$ .

$- 20 x - 4 x^{2} + 3 = 0$

$y = - 2 x - 1$

$- 20 x - 4 x^{2} + 3 = 0$

$y = - 2 x - 1$

Step 3.2

Use the quadratic formula to find the solutions.

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

$y = - 2 x - 1$

Step 3.3

Substitute the values $a = - 4$ , $b = - 20$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (- 4 \cdot 3)}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot - 4 \cdot 3}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4.1.2

Multiply $- 4 \cdot - 4 \cdot 3$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 \cdot 3}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4.1.2.2

Multiply $16$ by $3$ .

$x = \frac{20 \pm \sqrt{400 + 48}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm \sqrt{400 + 48}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4.1.3

Add $400$ and $48$ .

$x = \frac{20 \pm \sqrt{448}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4.1.4

Rewrite $448$ as $8^{2} \cdot 7$ .

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Step 3.4.1.4.1

Factor $64$ out of $448$ .

$x = \frac{20 \pm \sqrt{64 (7)}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{20 \pm \sqrt{8^{2} \cdot 7}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm \sqrt{8^{2} \cdot 7}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 8 \sqrt{7}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm 8 \sqrt{7}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.4.2

Multiply $2$ by $- 4$ .

$x = \frac{20 \pm 8 \sqrt{7}}{- 8}$

$y = - 2 x - 1$

Step 3.4.3

Simplify $\frac{20 \pm 8 \sqrt{7}}{- 8}$ .

$x = \frac{5 \pm 2 \sqrt{7}}{- 2}$

$y = - 2 x - 1$

Step 3.4.4

Move the negative in front of the fraction.

$x = - \frac{5 \pm 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

$x = - \frac{5 \pm 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

Step 3.5

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

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot - 4 \cdot 3}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.5.1.2

Multiply $- 4 \cdot - 4 \cdot 3$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 \cdot 3}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.5.1.2.2

Multiply $16$ by $3$ .

$x = \frac{20 \pm \sqrt{400 + 48}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm \sqrt{400 + 48}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.5.1.3

Add $400$ and $48$ .

$x = \frac{20 \pm \sqrt{448}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.5.1.4

Rewrite $448$ as $8^{2} \cdot 7$ .

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Step 3.5.1.4.1

Factor $64$ out of $448$ .

$x = \frac{20 \pm \sqrt{64 (7)}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.5.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{20 \pm \sqrt{8^{2} \cdot 7}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm \sqrt{8^{2} \cdot 7}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.5.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 8 \sqrt{7}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm 8 \sqrt{7}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.5.2

Multiply $2$ by $- 4$ .

$x = \frac{20 \pm 8 \sqrt{7}}{- 8}$

$y = - 2 x - 1$

Step 3.5.3

Simplify $\frac{20 \pm 8 \sqrt{7}}{- 8}$ .

$x = \frac{5 \pm 2 \sqrt{7}}{- 2}$

$y = - 2 x - 1$

Step 3.5.4

Move the negative in front of the fraction.

$x = - \frac{5 \pm 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

Step 3.5.5

Change the $\pm$ to $+$ .

$x = - \frac{5 + 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

Step 3.6

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

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot - 4 \cdot 3}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.6.1.2

Multiply $- 4 \cdot - 4 \cdot 3$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 \cdot 3}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.6.1.2.2

Multiply $16$ by $3$ .

$x = \frac{20 \pm \sqrt{400 + 48}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm \sqrt{400 + 48}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.6.1.3

Add $400$ and $48$ .

$x = \frac{20 \pm \sqrt{448}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.6.1.4

Rewrite $448$ as $8^{2} \cdot 7$ .

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Step 3.6.1.4.1

Factor $64$ out of $448$ .

$x = \frac{20 \pm \sqrt{64 (7)}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.6.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{20 \pm \sqrt{8^{2} \cdot 7}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm \sqrt{8^{2} \cdot 7}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 8 \sqrt{7}}{2 \cdot - 4}$

$y = - 2 x - 1$

$x = \frac{20 \pm 8 \sqrt{7}}{2 \cdot - 4}$

$y = - 2 x - 1$

Step 3.6.2

Multiply $2$ by $- 4$ .

$x = \frac{20 \pm 8 \sqrt{7}}{- 8}$

$y = - 2 x - 1$

Step 3.6.3

Simplify $\frac{20 \pm 8 \sqrt{7}}{- 8}$ .

$x = \frac{5 \pm 2 \sqrt{7}}{- 2}$

$y = - 2 x - 1$

Step 3.6.4

Move the negative in front of the fraction.

$x = - \frac{5 \pm 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

Step 3.6.5

Change the $\pm$ to $-$ .

$x = - \frac{5 - 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

Step 3.7

The final answer is the combination of both solutions.

$x = - \frac{5 + 2 \sqrt{7}}{2}, - \frac{5 - 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}, - \frac{5 - 2 \sqrt{7}}{2}$

$y = - 2 x - 1$

Step 4

Replace all occurrences of $x$ with $- \frac{5 + 2 \sqrt{7}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 2 x - 1$ with $- \frac{5 + 2 \sqrt{7}}{2}$ .

$y = - 2 (- \frac{5 + 2 \sqrt{7}}{2}) - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 2 (- \frac{5 + 2 \sqrt{7}}{2}) - 1$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.1

Move the leading negative in $- \frac{5 + 2 \sqrt{7}}{2}$ into the numerator.

$y = - 2 \frac{- (5 + 2 \sqrt{7})}{2} - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 4.2.1.1.1.2

Factor $2$ out of $- 2$ .

$y = 2 (- 1) (\frac{- (5 + 2 \sqrt{7})}{2}) - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 4.2.1.1.1.3

Cancel the common factor.

$y = 2 \cdot (- 1 \frac{- (5 + 2 \sqrt{7})}{2}) - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 4.2.1.1.1.4

Rewrite the expression.

$y = - 1 (- (5 + 2 \sqrt{7})) - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

$y = - 1 (- (5 + 2 \sqrt{7})) - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 4.2.1.1.2

Multiply $- 1$ by $- 1$ .

$y = 1 (5 + 2 \sqrt{7}) - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 4.2.1.1.3

Multiply $5 + 2 \sqrt{7}$ by $1$ .

$y = 5 + 2 \sqrt{7} - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

$y = 5 + 2 \sqrt{7} - 1$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 4.2.1.2

Subtract $1$ from $5$ .

$y = 4 + 2 \sqrt{7}$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

$y = 4 + 2 \sqrt{7}$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

$y = 4 + 2 \sqrt{7}$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

$y = 4 + 2 \sqrt{7}$

$x = - \frac{5 + 2 \sqrt{7}}{2}$

Step 5

Replace all occurrences of $x$ with $- \frac{5 - 2 \sqrt{7}}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 2 x - 1$ with $- \frac{5 - 2 \sqrt{7}}{2}$ .

$y = - 2 (- \frac{5 - 2 \sqrt{7}}{2}) - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 2 (- \frac{5 - 2 \sqrt{7}}{2}) - 1$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1.1

Move the leading negative in $- \frac{5 - 2 \sqrt{7}}{2}$ into the numerator.

$y = - 2 \frac{- (5 - 2 \sqrt{7})}{2} - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 5.2.1.1.1.2

Factor $2$ out of $- 2$ .

$y = 2 (- 1) (\frac{- (5 - 2 \sqrt{7})}{2}) - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 5.2.1.1.1.3

Cancel the common factor.

$y = 2 \cdot (- 1 \frac{- (5 - 2 \sqrt{7})}{2}) - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 5.2.1.1.1.4

Rewrite the expression.

$y = - 1 (- (5 - 2 \sqrt{7})) - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

$y = - 1 (- (5 - 2 \sqrt{7})) - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 5.2.1.1.2

Multiply $- 1$ by $- 1$ .

$y = 1 (5 - 2 \sqrt{7}) - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 5.2.1.1.3

Multiply $5 - 2 \sqrt{7}$ by $1$ .

$y = 5 - 2 \sqrt{7} - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

$y = 5 - 2 \sqrt{7} - 1$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 5.2.1.2

Subtract $1$ from $5$ .

$y = 4 - 2 \sqrt{7}$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

$y = 4 - 2 \sqrt{7}$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

$y = 4 - 2 \sqrt{7}$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

$y = 4 - 2 \sqrt{7}$

$x = - \frac{5 - 2 \sqrt{7}}{2}$

Step 6

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

$(- \frac{5 + 2 \sqrt{7}}{2}, 4 + 2 \sqrt{7})$

$(- \frac{5 - 2 \sqrt{7}}{2}, 4 - 2 \sqrt{7})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- \frac{5 + 2 \sqrt{7}}{2}, 4 + 2 \sqrt{7}), (- \frac{5 - 2 \sqrt{7}}{2}, 4 - 2 \sqrt{7})$

Equation Form:

$x = - \frac{5 + 2 \sqrt{7}}{2}, y = 4 + 2 \sqrt{7}$

$x = - \frac{5 - 2 \sqrt{7}}{2}, y = 4 - 2 \sqrt{7}$

Step 8