Find the x and y Intercepts y=x^3-125

$y = x^{3} - 125$

Step 1

Find the x-intercepts.

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

$0 = x^{3} - 125$

Solve the equation.

Rewrite the equation as $x^{3} - 125 = 0$ .

$x^{3} - 125 = 0$

Add $125$ to both sides of the equation.

$x^{3} = 125$

Subtract $125$ from both sides of the equation.

$x^{3} - 125 = 0$

Factor the left side of the equation.

Rewrite $125$ as $5^{3}$ .

$x^{3} - 5^{3} = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 5$ .

$(x - 5) (x^{2} + x \cdot 5 + 5^{2}) = 0$

Simplify.

Move $5$ to the left of $x$ .

$(x - 5) (x^{2} + 5 x + 5^{2}) = 0$

Raise $5$ to the power of $2$ .

$(x - 5) (x^{2} + 5 x + 25) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x^{2} + 5 x + 25 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x^{2} + 5 x + 25$ equal to $0$ and solve for $x$ .

Set $x^{2} + 5 x + 25$ equal to $0$ .

$x^{2} + 5 x + 25 = 0$

Solve $x^{2} + 5 x + 25 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 5$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$x = \frac{- 5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Subtract $100$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 75}}{2 \cdot 1}$

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

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

$x = \frac{- 5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$x = \frac{- 5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Move $5$ to the left of $i$ .

$x = \frac{- 5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm 5 i \sqrt{3}}{2}$

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

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$x = \frac{- 5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Subtract $100$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 75}}{2 \cdot 1}$

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

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

$x = \frac{- 5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$x = \frac{- 5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Move $5$ to the left of $i$ .

$x = \frac{- 5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm 5 i \sqrt{3}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 5 + 5 i \sqrt{3}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + 5 i \sqrt{3}}{2}$

Factor $- 1$ out of $5 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 5 - (- 5 i \sqrt{3})}{2}$

Factor $- 1$ out of $- 1 (5) - (- 5 i \sqrt{3})$ .

$x = \frac{- 1 (5 - 5 i \sqrt{3})}{2}$

Move the negative in front of the fraction.

$x = - \frac{5 - 5 i \sqrt{3}}{2}$

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

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$x = \frac{- 5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Subtract $100$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 75}}{2 \cdot 1}$

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

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

$x = \frac{- 5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$x = \frac{- 5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Move $5$ to the left of $i$ .

$x = \frac{- 5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm 5 i \sqrt{3}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 5 - 5 i \sqrt{3}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - 5 i \sqrt{3}}{2}$

Factor $- 1$ out of $- 5 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 5 - (5 i \sqrt{3})}{2}$

Factor $- 1$ out of $- 1 (5) - (5 i \sqrt{3})$ .

$x = \frac{- 1 (5 + 5 i \sqrt{3})}{2}$

Move the negative in front of the fraction.

$x = - \frac{5 + 5 i \sqrt{3}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{5 - 5 i \sqrt{3}}{2}, - \frac{5 + 5 i \sqrt{3}}{2}$

The final solution is all the values that make $(x - 5) (x^{2} + 5 x + 25) = 0$ true.

$x = 5, - \frac{5 - 5 i \sqrt{3}}{2}, - \frac{5 + 5 i \sqrt{3}}{2}$

x-intercept(s) in point form.

x-intercept(s): $(5, 0)$

Step 2

Find the y-intercepts.

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

$y = {(0)}^{3} - 125$

Solve the equation.

Remove parentheses.

$y = 0^{3} - 125$

Remove parentheses.

$y = {(0)}^{3} - 125$

Simplify ${(0)}^{3} - 125$ .

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

$y = 0 - 125$

Subtract $125$ from $0$ .

$y = - 125$

y-intercept(s) in point form.

y-intercept(s): $(0, - 125)$

Step 3

List the intersections.

x-intercept(s): $(5, 0)$

y-intercept(s): $(0, - 125)$

Step 4

Find the x and y Intercepts y=x^3-125

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