Algebra Examples

$2 x^{2} + 4 x - y = - 3$ $- 2 x + y = - 4$

Step 1

Add $2 x$ to both sides of the equation.

$y = - 4 + 2 x$

$2 x^{2} + 4 x - y = - 3$

Step 2

Replace all occurrences of $y$ with $- 4 + 2 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x^{2} + 4 x - y = - 3$ with $- 4 + 2 x$ .

$2 x^{2} + 4 x - (- 4 + 2 x) = - 3$

$y = - 4 + 2 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$2 x^{2} + 4 x + 4 - (2 x) = - 3$

$y = - 4 + 2 x$

Step 2.2.1.1.2

Multiply $- 1$ by $- 4$ .

$2 x^{2} + 4 x + 4 - (2 x) = - 3$

$y = - 4 + 2 x$

Step 2.2.1.1.3

Multiply $2$ by $- 1$ .

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

$y = - 4 + 2 x$

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

$y = - 4 + 2 x$

Step 2.2.1.2

Subtract $2 x$ from $4 x$ .

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

$y = - 4 + 2 x$

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

$y = - 4 + 2 x$

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

$y = - 4 + 2 x$

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

$y = - 4 + 2 x$

Step 3

Solve for $x$ in $2 x^{2} + 2 x + 4 = - 3$ .

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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 $3$ to both sides of the equation.

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

$y = - 4 + 2 x$

Step 3.1.2

Add $4$ and $3$ .

$2 x^{2} + 2 x + 7 = 0$

$y = - 4 + 2 x$

$2 x^{2} + 2 x + 7 = 0$

$y = - 4 + 2 x$

Step 3.2

Use the quadratic formula to find the solutions.

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

$y = - 4 + 2 x$

Step 3.3

Substitute the values $a = 2$ , $b = 2$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (2 \cdot 7)}}{2 \cdot 2}$

$y = - 4 + 2 x$

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 $2$ to the power of $2$ .

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

$y = - 4 + 2 x$

Step 3.4.1.2

Multiply $- 4 \cdot 2 \cdot 7$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $2$ .

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

$y = - 4 + 2 x$

Step 3.4.1.2.2

Multiply $- 8$ by $7$ .

$x = \frac{- 2 \pm \sqrt{4 - 56}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm \sqrt{4 - 56}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.3

Subtract $56$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.4

Rewrite $- 52$ as $- 1 (52)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (52)}$ as $\sqrt{- 1} \cdot \sqrt{52}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.7

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 3.4.1.7.1

Factor $4$ out of $52$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (13)}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 13}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 13}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{13})}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{13}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm 2 i \sqrt{13}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.4.2

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 i \sqrt{13}}{4}$

$y = - 4 + 2 x$

Step 3.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{13}}{4}$ .

$x = \frac{- 1 \pm i \sqrt{13}}{2}$

$y = - 4 + 2 x$

$x = \frac{- 1 \pm i \sqrt{13}}{2}$

$y = - 4 + 2 x$

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 $2$ to the power of $2$ .

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

$y = - 4 + 2 x$

Step 3.5.1.2

Multiply $- 4 \cdot 2 \cdot 7$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $2$ .

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

$y = - 4 + 2 x$

Step 3.5.1.2.2

Multiply $- 8$ by $7$ .

$x = \frac{- 2 \pm \sqrt{4 - 56}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm \sqrt{4 - 56}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.3

Subtract $56$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.4

Rewrite $- 52$ as $- 1 (52)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (52)}$ as $\sqrt{- 1} \cdot \sqrt{52}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.7

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 3.5.1.7.1

Factor $4$ out of $52$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (13)}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 13}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 13}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{13})}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{13}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm 2 i \sqrt{13}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.5.2

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 i \sqrt{13}}{4}$

$y = - 4 + 2 x$

Step 3.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{13}}{4}$ .

$x = \frac{- 1 \pm i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 3.5.4

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 3.5.5

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

$x = \frac{- 1 \cdot 1 + i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 3.5.6

Factor $- 1$ out of $i \sqrt{13}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{13})}{2}$

$y = - 4 + 2 x$

Step 3.5.7

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{13})$ .

$x = \frac{- 1 (1 - i \sqrt{13})}{2}$

$y = - 4 + 2 x$

Step 3.5.8

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 4 + 2 x$

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 4 + 2 x$

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 $2$ to the power of $2$ .

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

$y = - 4 + 2 x$

Step 3.6.1.2

Multiply $- 4 \cdot 2 \cdot 7$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $2$ .

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

$y = - 4 + 2 x$

Step 3.6.1.2.2

Multiply $- 8$ by $7$ .

$x = \frac{- 2 \pm \sqrt{4 - 56}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm \sqrt{4 - 56}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.3

Subtract $56$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.4

Rewrite $- 52$ as $- 1 (52)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.5

Rewrite $\sqrt{- 1 (52)}$ as $\sqrt{- 1} \cdot \sqrt{52}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{52}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.7

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 3.6.1.7.1

Factor $4$ out of $52$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (13)}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 13}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 13}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{13})}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{13}}{2 \cdot 2}$

$y = - 4 + 2 x$

$x = \frac{- 2 \pm 2 i \sqrt{13}}{2 \cdot 2}$

$y = - 4 + 2 x$

Step 3.6.2

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 i \sqrt{13}}{4}$

$y = - 4 + 2 x$

Step 3.6.3

Simplify $\frac{- 2 \pm 2 i \sqrt{13}}{4}$ .

$x = \frac{- 1 \pm i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 3.6.4

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 3.6.5

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

$x = \frac{- 1 \cdot 1 - i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 3.6.6

Factor $- 1$ out of $- i \sqrt{13}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{13})}{2}$

$y = - 4 + 2 x$

Step 3.6.7

Factor $- 1$ out of $- 1 (1) - (i \sqrt{13})$ .

$x = \frac{- 1 (1 + i \sqrt{13})}{2}$

$y = - 4 + 2 x$

Step 3.6.8

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{13}}{2}$

$y = - 4 + 2 x$

$x = - \frac{1 + i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 3.7

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{13}}{2}, - \frac{1 + i \sqrt{13}}{2}$

$y = - 4 + 2 x$

$x = - \frac{1 - i \sqrt{13}}{2}, - \frac{1 + i \sqrt{13}}{2}$

$y = - 4 + 2 x$

Step 4

Replace all occurrences of $x$ with $- \frac{1 - i \sqrt{13}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 4 + 2 x$ with $- \frac{1 - i \sqrt{13}}{2}$ .

$y = - 4 + 2 (- \frac{1 - i \sqrt{13}}{2})$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 4 + 2 (- \frac{1 - i \sqrt{13}}{2})$ .

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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 $- \frac{1 - i \sqrt{13}}{2}$ into the numerator.

$y = - 4 + 2 (\frac{- (1 - i \sqrt{13})}{2})$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2.1.1.1.2

Cancel the common factor.

$y = - 4 + 2 (\frac{- (1 - i \sqrt{13})}{2})$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2.1.1.1.3

Rewrite the expression.

$y = - 4 - (1 - i \sqrt{13})$

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 4 - (1 - i \sqrt{13})$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2.1.1.2

Apply the distributive property.

$y = - 4 - 1 \cdot 1 - (- i \sqrt{13})$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2.1.1.3

Multiply $- 1$ by $1$ .

$y = - 4 - 1 - (- i \sqrt{13})$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2.1.1.4

Multiply $- (- i \sqrt{13})$ .

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Step 4.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$y = - 4 - 1 + 1 (i \sqrt{13})$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2.1.1.4.2

Multiply $\sqrt{13}$ by $1$ .

$y = - 4 - 1 + \sqrt{13} i$

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 4 - 1 + \sqrt{13} i$

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 4 - 1 + \sqrt{13} i$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 4.2.1.2

Subtract $1$ from $- 4$ .

$y = - 5 + \sqrt{13} i$

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 5 + \sqrt{13} i$

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 5 + \sqrt{13} i$

$x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 5 + \sqrt{13} i$

$x = - \frac{1 - i \sqrt{13}}{2}$

Step 5

Replace all occurrences of $x$ with $- \frac{1 + i \sqrt{13}}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 4 + 2 x$ with $- \frac{1 + i \sqrt{13}}{2}$ .

$y = - 4 + 2 (- \frac{1 + i \sqrt{13}}{2})$

$x = - \frac{1 + i \sqrt{13}}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 4 + 2 (- \frac{1 + i \sqrt{13}}{2})$ .

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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 $- \frac{1 + i \sqrt{13}}{2}$ into the numerator.

$y = - 4 + 2 (\frac{- (1 + i \sqrt{13})}{2})$

$x = - \frac{1 + i \sqrt{13}}{2}$

Step 5.2.1.1.1.2

Cancel the common factor.

$y = - 4 + 2 (\frac{- (1 + i \sqrt{13})}{2})$

$x = - \frac{1 + i \sqrt{13}}{2}$

Step 5.2.1.1.1.3

Rewrite the expression.

$y = - 4 - (1 + i \sqrt{13})$

$x = - \frac{1 + i \sqrt{13}}{2}$

$y = - 4 - (1 + i \sqrt{13})$

$x = - \frac{1 + i \sqrt{13}}{2}$

Step 5.2.1.1.2

Apply the distributive property.

$y = - 4 - 1 \cdot 1 - (i \sqrt{13})$

$x = - \frac{1 + i \sqrt{13}}{2}$

Step 5.2.1.1.3

Multiply $- 1$ by $1$ .

$y = - 4 - 1 - i \sqrt{13}$

$x = - \frac{1 + i \sqrt{13}}{2}$

$y = - 4 - 1 - i \sqrt{13}$

$x = - \frac{1 + i \sqrt{13}}{2}$

Step 5.2.1.2

Subtract $1$ from $- 4$ .

$y = - 5 - i \sqrt{13}$

$x = - \frac{1 + i \sqrt{13}}{2}$

$y = - 5 - i \sqrt{13}$

$x = - \frac{1 + i \sqrt{13}}{2}$

$y = - 5 - i \sqrt{13}$

$x = - \frac{1 + i \sqrt{13}}{2}$

$y = - 5 - i \sqrt{13}$

$x = - \frac{1 + i \sqrt{13}}{2}$

Step 6

List all of the solutions.

$y = - 5 + \sqrt{13} i, x = - \frac{1 - i \sqrt{13}}{2}$

$y = - 5 - i \sqrt{13}, x = - \frac{1 + i \sqrt{13}}{2}$

Step 7