Find the x and y Intercepts f(x)=2(5x^2-2)(x-5)(4x^2-2)

$f (x) = 2 (5 x^{2} - 2) (x - 5) (4 x^{2} - 2)$

Step 1

Find the x-intercepts.

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

$0 = 2 (5 x^{2} - 2) (x - 5) (4 x^{2} - 2)$

Solve the equation.

Rewrite the equation as $2 (5 x^{2} - 2) (x - 5) (4 x^{2} - 2) = 0$ .

$2 (5 x^{2} - 2) (x - 5) (4 x^{2} - 2) = 0$

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

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

$x - 5 = 0$

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

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

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

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

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

Add $2$ to both sides of the equation.

$5 x^{2} = 2$

Divide each term in $5 x^{2} = 2$ by $5$ and simplify.

Divide each term in $5 x^{2} = 2$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{2}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{2}{5}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{2}{5}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{2}{5}}$

Simplify $\pm \sqrt{\frac{2}{5}}$ .

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{5}}$

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5}$

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $5$ .

$x = \pm \frac{\sqrt{10}}{5}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{10}}{5}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{10}}{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{10}}{5}, - \frac{\sqrt{10}}{5}$

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 $4 x^{2} - 2$ equal to $0$ and solve for $x$ .

Set $4 x^{2} - 2$ equal to $0$ .

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

Solve $4 x^{2} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$4 x^{2} = 2$

Divide each term in $4 x^{2} = 2$ by $4$ and simplify.

Divide each term in $4 x^{2} = 2$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{2}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{2}{4}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{2}{4}$

Simplify the right side.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$x^{2} = \frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x^{2} = \frac{2 \cdot 1}{2 \cdot 2}$

Cancel the common factor.

$x^{2} = \frac{2 \cdot 1}{2 \cdot 2}$

Rewrite the expression.

$x^{2} = \frac{1}{2}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{2}}$

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

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

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

Combine and simplify the denominator.

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

$x = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{2}}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

The final solution is all the values that make $2 (5 x^{2} - 2) (x - 5) (4 x^{2} - 2) = 0$ true.

$x = \frac{\sqrt{10}}{5}, - \frac{\sqrt{10}}{5}, 5, \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

x-intercept(s) in point form.

x-intercept(s): $(\frac{\sqrt{10}}{5}, 0), (- \frac{\sqrt{10}}{5}, 0), (5, 0), (\frac{\sqrt{2}}{2}, 0), (- \frac{\sqrt{2}}{2}, 0)$

Step 2

Find the y-intercepts.

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

$y = 2 (5 {(0)}^{2} - 2) ((0) - 5) (4 {(0)}^{2} - 2)$

Solve the equation.

Remove parentheses.

$y = 2 (5 \cdot 0^{2} - 2) ((0) - 5) (4 {(0)}^{2} - 2)$

Remove parentheses.

$y = 2 (5 \cdot 0^{2} - 2) (0 - 5) (4 {(0)}^{2} - 2)$

Remove parentheses.

$y = 2 (5 \cdot 0^{2} - 2) (0 - 5) (4 \cdot 0^{2} - 2)$

Remove parentheses.

$y = 2 (5 {(0)}^{2} - 2) ((0) - 5) (4 {(0)}^{2} - 2)$

Simplify $2 (5 {(0)}^{2} - 2) ((0) - 5) (4 {(0)}^{2} - 2)$ .

Simplify each term.

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

$y = 2 (5 \cdot 0 - 2) ((0) - 5) (4 {(0)}^{2} - 2)$

Multiply $5$ by $0$ .

$y = 2 (0 - 2) ((0) - 5) (4 {(0)}^{2} - 2)$

Simplify the expression.

Subtract $2$ from $0$ .

$y = 2 \cdot - 2 ((0) - 5) (4 {(0)}^{2} - 2)$

Multiply $2$ by $- 2$ .

$y = - 4 ((0) - 5) (4 {(0)}^{2} - 2)$

Subtract $5$ from $0$ .

$y = - 4 \cdot - 5 (4 {(0)}^{2} - 2)$

Multiply $- 4$ by $- 5$ .

$y = 20 (4 {(0)}^{2} - 2)$

Simplify each term.

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

$y = 20 (4 \cdot 0 - 2)$

Multiply $4$ by $0$ .

$y = 20 (0 - 2)$

Simplify the expression.

Subtract $2$ from $0$ .

$y = 20 \cdot - 2$

Multiply $20$ by $- 2$ .

$y = - 40$

y-intercept(s) in point form.

y-intercept(s): $(0, - 40)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\sqrt{10}}{5}, 0), (- \frac{\sqrt{10}}{5}, 0), (5, 0), (\frac{\sqrt{2}}{2}, 0), (- \frac{\sqrt{2}}{2}, 0)$

y-intercept(s): $(0, - 40)$

Step 4

Find the x and y Intercepts f(x)=2(5x^2-2)(x-5)(4x^2-2)

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