Solve the System of Equations y=-2x^2-4x-1 y=2x+4

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

Step 1

Eliminate the equal sides of each equation and combine.

$- 2 x^{2} - 4 x - 1 = 2 x + 4$

Step 2

Solve $- 2 x^{2} - 4 x - 1 = 2 x + 4$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

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

$- 2 x^{2} - 4 x - 1 - 2 x = 4$

Subtract $2 x$ from $- 4 x$ .

$- 2 x^{2} - 6 x - 1 = 4$

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

Subtract $4$ from both sides of the equation.

$- 2 x^{2} - 6 x - 1 - 4 = 0$

Subtract $4$ from $- 1$ .

$- 2 x^{2} - 6 x - 5 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 2$ , $b = - 6$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (- 2 \cdot - 5)}}{2 \cdot - 2} + y = 2 x + 4$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 2 \cdot - 5}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 5$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 8 \cdot - 5}}{2 \cdot - 2}$

Multiply $8$ by $- 5$ .

$x = \frac{6 \pm \sqrt{36 - 40}}{2 \cdot - 2}$

Subtract $40$ from $36$ .

$x = \frac{6 \pm \sqrt{- 4}}{2 \cdot - 2}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot - 2}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot \sqrt{4}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot 2}{2 \cdot - 2}$

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

$x = \frac{6 \pm 2 i}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{6 \pm 2 i}{- 4}$

Simplify $\frac{6 \pm 2 i}{- 4}$ .

$x = \frac{3 \pm i}{- 2}$

Move the negative in front of the fraction.

$x = - \frac{3 \pm i}{2}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 2 \cdot - 5}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 5$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 8 \cdot - 5}}{2 \cdot - 2}$

Multiply $8$ by $- 5$ .

$x = \frac{6 \pm \sqrt{36 - 40}}{2 \cdot - 2}$

Subtract $40$ from $36$ .

$x = \frac{6 \pm \sqrt{- 4}}{2 \cdot - 2}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot - 2}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot \sqrt{4}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot 2}{2 \cdot - 2}$

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

$x = \frac{6 \pm 2 i}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{6 \pm 2 i}{- 4}$

Simplify $\frac{6 \pm 2 i}{- 4}$ .

$x = \frac{3 \pm i}{- 2}$

Move the negative in front of the fraction.

$x = - \frac{3 \pm i}{2}$

Change the $\pm$ to $+$ .

$x = - \frac{3 + i}{2}$

Split the fraction $\frac{3 + i}{2}$ into two fractions.

$x = - (\frac{3}{2} + \frac{i}{2})$

Apply the distributive property.

$x = - \frac{3}{2} - \frac{i}{2}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 2 \cdot - 5}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 5$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 8 \cdot - 5}}{2 \cdot - 2}$

Multiply $8$ by $- 5$ .

$x = \frac{6 \pm \sqrt{36 - 40}}{2 \cdot - 2}$

Subtract $40$ from $36$ .

$x = \frac{6 \pm \sqrt{- 4}}{2 \cdot - 2}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot - 2}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot \sqrt{4}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot - 2}$

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

$x = \frac{6 \pm i \cdot 2}{2 \cdot - 2}$

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

$x = \frac{6 \pm 2 i}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{6 \pm 2 i}{- 4}$

Simplify $\frac{6 \pm 2 i}{- 4}$ .

$x = \frac{3 \pm i}{- 2}$

Move the negative in front of the fraction.

$x = - \frac{3 \pm i}{2}$

Change the $\pm$ to $-$ .

$x = - \frac{3 - i}{2}$

Split the fraction $\frac{3 - i}{2}$ into two fractions.

$x = - (\frac{3}{2} + \frac{- i}{2})$

Move the negative in front of the fraction.

$x = - (\frac{3}{2} - \frac{i}{2})$

Apply the distributive property.

$x = - \frac{3}{2} + \frac{i}{2}$

Multiply $- - \frac{i}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = - \frac{3}{2} + 1 (\frac{i}{2})$

Multiply $\frac{i}{2}$ by $1$ .

$x = - \frac{3}{2} + \frac{i}{2}$

The final answer is the combination of both solutions.

$x = - \frac{3}{2} - \frac{i}{2}, - \frac{3}{2} + \frac{i}{2}$

Step 3

Evaluate $y$ when $x = - \frac{3}{2} - \frac{i}{2}$ .

Substitute $- \frac{3}{2} - \frac{i}{2}$ for $x$ .

$y = 2 (- \frac{3}{2} - \frac{i}{2}) + 4$

Simplify $2 (- \frac{3}{2} - \frac{i}{2}) + 4$ .

Simplify each term.

Apply the distributive property.

$y = 2 (- \frac{3}{2}) + 2 (- \frac{i}{2}) + 4$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y = 2 (\frac{- 3}{2}) + 2 (- \frac{i}{2}) + 4$

Cancel the common factor.

$y = 2 (\frac{- 3}{2}) + 2 (- \frac{i}{2}) + 4$

Rewrite the expression.

$y = - 3 + 2 (- \frac{i}{2}) + 4$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{i}{2}$ into the numerator.

$y = - 3 + 2 \frac{- i}{2} + 4$

Cancel the common factor.

$y = - 3 + 2 \frac{- i}{2} + 4$

Rewrite the expression.

$y = - 3 - i + 4$

Add $- 3$ and $4$ .

$y = 1 - i$

Step 4

Evaluate $y$ when $x = - \frac{3}{2} + \frac{i}{2}$ .

Substitute $- \frac{3}{2} + \frac{i}{2}$ for $x$ .

$y = 2 (- \frac{3}{2} + \frac{i}{2}) + 4$

Simplify $2 (- \frac{3}{2} + \frac{i}{2}) + 4$ .

Simplify each term.

Apply the distributive property.

$y = 2 (- \frac{3}{2}) + 2 \frac{i}{2} + 4$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y = 2 (\frac{- 3}{2}) + 2 \frac{i}{2} + 4$

Cancel the common factor.

$y = 2 (\frac{- 3}{2}) + 2 \frac{i}{2} + 4$

Rewrite the expression.

$y = - 3 + 2 \frac{i}{2} + 4$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = - 3 + 2 \frac{i}{2} + 4$

Rewrite the expression.

$y = - 3 + i + 4$

Add $- 3$ and $4$ .

$y = 1 + i$

Step 5

List all of the solutions.

$y = 1 - i, x = - \frac{3}{2} - \frac{i}{2}$

$y = 1 + i, x = - \frac{3}{2} + \frac{i}{2}$

Step 6

Solve the System of Equations y=-2x^2-4x-1 y=2x+4

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