Find the x and y Intercepts 2x^2-20x+57

$2 x^{2} - 20 x + 57$

Step 1

Write $2 x^{2} - 20 x + 57$ as an equation.

$y = 2 x^{2} - 20 x + 57$

Step 2

Find the x-intercepts.

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

$0 = 2 x^{2} - 20 x + 57$

Solve the equation.

Rewrite the equation as $2 x^{2} - 20 x + 57 = 0$ .

$2 x^{2} - 20 x + 57 = 0$

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 $c = 57$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (2 \cdot 57)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

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

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 2 \cdot 57}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 57$ .

Multiply $- 4$ by $2$ .

$x = \frac{20 \pm \sqrt{400 - 8 \cdot 57}}{2 \cdot 2}$

Multiply $- 8$ by $57$ .

$x = \frac{20 \pm \sqrt{400 - 456}}{2 \cdot 2}$

Subtract $456$ from $400$ .

$x = \frac{20 \pm \sqrt{- 56}}{2 \cdot 2}$

Rewrite $- 56$ as $- 1 (56)$ .

$x = \frac{20 \pm \sqrt{- 1 \cdot 56}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$x = \frac{20 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot 2}$

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

$x = \frac{20 \pm i \cdot \sqrt{56}}{2 \cdot 2}$

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$x = \frac{20 \pm i \cdot \sqrt{4 (14)}}{2 \cdot 2}$

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

$x = \frac{20 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{20 \pm i \cdot (2 \sqrt{14})}{2 \cdot 2}$

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

$x = \frac{20 \pm 2 i \sqrt{14}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{20 \pm 2 i \sqrt{14}}{4}$

Simplify $\frac{20 \pm 2 i \sqrt{14}}{4}$ .

$x = \frac{10 \pm i \sqrt{14}}{2}$

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

Simplify the numerator.

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

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 2 \cdot 57}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 57$ .

Multiply $- 4$ by $2$ .

$x = \frac{20 \pm \sqrt{400 - 8 \cdot 57}}{2 \cdot 2}$

Multiply $- 8$ by $57$ .

$x = \frac{20 \pm \sqrt{400 - 456}}{2 \cdot 2}$

Subtract $456$ from $400$ .

$x = \frac{20 \pm \sqrt{- 56}}{2 \cdot 2}$

Rewrite $- 56$ as $- 1 (56)$ .

$x = \frac{20 \pm \sqrt{- 1 \cdot 56}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$x = \frac{20 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot 2}$

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

$x = \frac{20 \pm i \cdot \sqrt{56}}{2 \cdot 2}$

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$x = \frac{20 \pm i \cdot \sqrt{4 (14)}}{2 \cdot 2}$

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

$x = \frac{20 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{20 \pm i \cdot (2 \sqrt{14})}{2 \cdot 2}$

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

$x = \frac{20 \pm 2 i \sqrt{14}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{20 \pm 2 i \sqrt{14}}{4}$

Simplify $\frac{20 \pm 2 i \sqrt{14}}{4}$ .

$x = \frac{10 \pm i \sqrt{14}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{10 + i \sqrt{14}}{2}$

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

Simplify the numerator.

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

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 2 \cdot 57}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 57$ .

Multiply $- 4$ by $2$ .

$x = \frac{20 \pm \sqrt{400 - 8 \cdot 57}}{2 \cdot 2}$

Multiply $- 8$ by $57$ .

$x = \frac{20 \pm \sqrt{400 - 456}}{2 \cdot 2}$

Subtract $456$ from $400$ .

$x = \frac{20 \pm \sqrt{- 56}}{2 \cdot 2}$

Rewrite $- 56$ as $- 1 (56)$ .

$x = \frac{20 \pm \sqrt{- 1 \cdot 56}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$x = \frac{20 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot 2}$

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

$x = \frac{20 \pm i \cdot \sqrt{56}}{2 \cdot 2}$

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$x = \frac{20 \pm i \cdot \sqrt{4 (14)}}{2 \cdot 2}$

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

$x = \frac{20 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{20 \pm i \cdot (2 \sqrt{14})}{2 \cdot 2}$

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

$x = \frac{20 \pm 2 i \sqrt{14}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{20 \pm 2 i \sqrt{14}}{4}$

Simplify $\frac{20 \pm 2 i \sqrt{14}}{4}$ .

$x = \frac{10 \pm i \sqrt{14}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{10 - i \sqrt{14}}{2}$

The final answer is the combination of both solutions.

$x = \frac{10 + i \sqrt{14}}{2}, \frac{10 - i \sqrt{14}}{2}$

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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

$y = 2 {(0)}^{2} - 20 \cdot 0 + 57$

Solve the equation.

Remove parentheses.

$y = 2 \cdot 0^{2} - 20 \cdot 0 + 57$

Remove parentheses.

$y = 2 {(0)}^{2} - 20 \cdot 0 + 57$

Simplify $2 {(0)}^{2} - 20 \cdot 0 + 57$ .

Simplify each term.

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

$y = 2 \cdot 0 - 20 \cdot 0 + 57$

Multiply $2$ by $0$ .

$y = 0 - 20 \cdot 0 + 57$

Multiply $- 20$ by $0$ .

$y = 0 + 0 + 57$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 57$

Add $0$ and $57$ .

$y = 57$

y-intercept(s) in point form.

y-intercept(s): $(0, 57)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 57)$

Step 5

Find the x and y Intercepts 2x^2-20x+57

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