Find the x and y Intercepts 2y^2+x+20y+51=0

$2 y^{2} + x + 20 y + 51 = 0$

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$2 {(0)}^{2} + x + 20 (0) + 51 = 0$

Solve the equation.

Simplify $2 {(0)}^{2} + x + 20 (0) + 51$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$2 \cdot 0 + x + 20 (0) + 51 = 0$

Multiply $2$ by $0$ .

$0 + x + 20 (0) + 51 = 0$

Multiply $20$ by $0$ .

$0 + x + 0 + 51 = 0$

Combine the opposite terms in $0 + x + 0 + 51$ .

Add $0$ and $x$ .

$x + 0 + 51 = 0$

Add $x$ and $0$ .

$x + 51 = 0$

Subtract $51$ from both sides of the equation.

$x = - 51$

x-intercept(s) in point form.

x-intercept(s): $(- 51, 0)$

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$2 y^{2} + 0 + 20 y + 51 = 0$

Solve the equation.

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

$2 y^{2} + 20 y + 51 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = 20$ , and $c = 51$ into the quadratic formula and solve for $y$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (2 \cdot 51)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

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

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot 2 \cdot 51}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$y = \frac{- 20 \pm \sqrt{400 - 8 \cdot 51}}{2 \cdot 2}$

Multiply $- 8$ by $51$ .

$y = \frac{- 20 \pm \sqrt{400 - 408}}{2 \cdot 2}$

Subtract $408$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 8}}{2 \cdot 2}$

Rewrite $- 8$ as $- 1 (8)$ .

$y = \frac{- 20 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$y = \frac{- 20 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 2}$

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

$y = \frac{- 20 \pm i \cdot \sqrt{8}}{2 \cdot 2}$

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

Factor $4$ out of $8$ .

$y = \frac{- 20 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 2}$

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

$y = \frac{- 20 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 20 \pm i \cdot (2 \sqrt{2})}{2 \cdot 2}$

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

$y = \frac{- 20 \pm 2 i \sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 20 \pm 2 i \sqrt{2}}{4}$

Simplify $\frac{- 20 \pm 2 i \sqrt{2}}{4}$ .

$y = \frac{- 10 \pm i \sqrt{2}}{2}$

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

Simplify the numerator.

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

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot 2 \cdot 51}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$y = \frac{- 20 \pm \sqrt{400 - 8 \cdot 51}}{2 \cdot 2}$

Multiply $- 8$ by $51$ .

$y = \frac{- 20 \pm \sqrt{400 - 408}}{2 \cdot 2}$

Subtract $408$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 8}}{2 \cdot 2}$

Rewrite $- 8$ as $- 1 (8)$ .

$y = \frac{- 20 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$y = \frac{- 20 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 2}$

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

$y = \frac{- 20 \pm i \cdot \sqrt{8}}{2 \cdot 2}$

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

Factor $4$ out of $8$ .

$y = \frac{- 20 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 2}$

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

$y = \frac{- 20 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 20 \pm i \cdot (2 \sqrt{2})}{2 \cdot 2}$

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

$y = \frac{- 20 \pm 2 i \sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 20 \pm 2 i \sqrt{2}}{4}$

Simplify $\frac{- 20 \pm 2 i \sqrt{2}}{4}$ .

$y = \frac{- 10 \pm i \sqrt{2}}{2}$

Change the $\pm$ to $+$ .

$y = \frac{- 10 + i \sqrt{2}}{2}$

Rewrite $- 10$ as $- 1 (10)$ .

$y = \frac{- 1 \cdot 10 + i \sqrt{2}}{2}$

Factor $- 1$ out of $i \sqrt{2}$ .

$y = \frac{- 1 \cdot 10 - (- i \sqrt{2})}{2}$

Factor $- 1$ out of $- 1 (10) - (- i \sqrt{2})$ .

$y = \frac{- 1 (10 - i \sqrt{2})}{2}$

Move the negative in front of the fraction.

$y = - \frac{10 - i \sqrt{2}}{2}$

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

Simplify the numerator.

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

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot 2 \cdot 51}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$y = \frac{- 20 \pm \sqrt{400 - 8 \cdot 51}}{2 \cdot 2}$

Multiply $- 8$ by $51$ .

$y = \frac{- 20 \pm \sqrt{400 - 408}}{2 \cdot 2}$

Subtract $408$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 8}}{2 \cdot 2}$

Rewrite $- 8$ as $- 1 (8)$ .

$y = \frac{- 20 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$y = \frac{- 20 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 2}$

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

$y = \frac{- 20 \pm i \cdot \sqrt{8}}{2 \cdot 2}$

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

Factor $4$ out of $8$ .

$y = \frac{- 20 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 2}$

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

$y = \frac{- 20 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 20 \pm i \cdot (2 \sqrt{2})}{2 \cdot 2}$

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

$y = \frac{- 20 \pm 2 i \sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 20 \pm 2 i \sqrt{2}}{4}$

Simplify $\frac{- 20 \pm 2 i \sqrt{2}}{4}$ .

$y = \frac{- 10 \pm i \sqrt{2}}{2}$

Change the $\pm$ to $-$ .

$y = \frac{- 10 - i \sqrt{2}}{2}$

Rewrite $- 10$ as $- 1 (10)$ .

$y = \frac{- 1 \cdot 10 - i \sqrt{2}}{2}$

Factor $- 1$ out of $- i \sqrt{2}$ .

$y = \frac{- 1 \cdot 10 - (i \sqrt{2})}{2}$

Factor $- 1$ out of $- 1 (10) - (i \sqrt{2})$ .

$y = \frac{- 1 (10 + i \sqrt{2})}{2}$

Move the negative in front of the fraction.

$y = - \frac{10 + i \sqrt{2}}{2}$

The final answer is the combination of both solutions.

$y = - \frac{10 - i \sqrt{2}}{2}, - \frac{10 + i \sqrt{2}}{2}$

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

List the intersections.

x-intercept(s): $(- 51, 0)$

y-intercept(s): $None$

Find the x and y Intercepts 2y^2+x+20y+51=0

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