Find the Roots (Zeros) 5x^4-40x^3+75x^2

$5 x^{4} - 40 x^{3} + 75 x^{2}$

Step 1

Set $5 x^{4} - 40 x^{3} + 75 x^{2}$ equal to $0$ .

$5 x^{4} - 40 x^{3} + 75 x^{2} = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $5 x^{2}$ out of $5 x^{4} - 40 x^{3} + 75 x^{2}$ .

Factor $5 x^{2}$ out of $5 x^{4}$ .

$5 x^{2} (x^{2}) - 40 x^{3} + 75 x^{2} = 0$

Factor $5 x^{2}$ out of $- 40 x^{3}$ .

$5 x^{2} (x^{2}) + 5 x^{2} (- 8 x) + 75 x^{2} = 0$

Factor $5 x^{2}$ out of $75 x^{2}$ .

$5 x^{2} (x^{2}) + 5 x^{2} (- 8 x) + 5 x^{2} (15) = 0$

Factor $5 x^{2}$ out of $5 x^{2} (x^{2}) + 5 x^{2} (- 8 x)$ .

$5 x^{2} (x^{2} - 8 x) + 5 x^{2} (15) = 0$

Factor $5 x^{2}$ out of $5 x^{2} (x^{2} - 8 x) + 5 x^{2} (15)$ .

$5 x^{2} (x^{2} - 8 x + 15) = 0$

Factor.

Factor $x^{2} - 8 x + 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$5 x^{2} ((x - 5) (x - 3)) = 0$

Remove unnecessary parentheses.

$5 x^{2} (x - 5) (x - 3) = 0$

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

$x^{2} = 0$

$x - 5 = 0$

$x - 3 = 0$

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

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 $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $5 x^{2} (x - 5) (x - 3) = 0$ true.

$x = 0, 5, 3$

Step 3

Find the Roots (Zeros) 5x^4-40x^3+75x^2

Related Questions

Support LCMGCF.com by disabling your adblocker.