QuestionFind the zeros of the function $f(x)=-25 x^{4}+4 x^{2}$ . Enter answers as a comma-separated list.

Studdy Solution

STEP 1

Assumptions1. The function is given by $f(x)=-25x^{4}+4x^{}$ . We need to find the zeros of the function, which are the values of $x$ that make $f(x)=0$

STEP 2

To find the zeros of the function, we need to set the function equal to zero and solve for $x$ .
$-25x^{4}+4x^{2}=0$

STEP 3

We can simplify this equation by dividing all terms by -1.
$25x^{}-x^{2}=0$

STEP 4

This is a quadratic equation in the form of $ax^{4}+bx^{2}+c=0$ , where $a=25$ , $b=-4$ , and $c=0$ . To solve this, we can use the substitution $y=x^{2}$ to turn it into a quadratic equation in $y$ .
$25y^{2}-4y=0$

STEP 5

This equation can be factored as follows $y(25y-4)=0$

STEP 6

Setting each factor equal to zero gives the solutions for $y$ :
$y=0, \quad y=\frac{4}{25}$

STEP 7

Substitute $x^{2}$ back in for $y$ to find the solutions for $x$ :
$x^{2}=0, \quad x^{2}=\frac{4}{25}$

STEP 8

olving these equations gives the zeros of the function $x=0, \quad x=\pm\frac{2}{5}$ The zeros of the function are $x=0$ , $x=\frac{2}{5}$ , and $x=-\frac{2}{5}$ .

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QuestionFind the zeros of the function $f(x)=-25 x^{4}+4 x^{2}$ . Enter answers as a comma-separated list.

Studdy Solution

STEP 1

Assumptions1. The function is given by $f(x)=-25x^{4}+4x^{}$ . We need to find the zeros of the function, which are the values of $x$ that make $f(x)=0$

STEP 2

To find the zeros of the function, we need to set the function equal to zero and solve for $x$ .
$-25x^{4}+4x^{2}=0$

STEP 3

We can simplify this equation by dividing all terms by -1.
$25x^{}-x^{2}=0$

STEP 4

This is a quadratic equation in the form of $ax^{4}+bx^{2}+c=0$ , where $a=25$ , $b=-4$ , and $c=0$ . To solve this, we can use the substitution $y=x^{2}$ to turn it into a quadratic equation in $y$ .
$25y^{2}-4y=0$

STEP 5

This equation can be factored as follows $y(25y-4)=0$

STEP 6

Setting each factor equal to zero gives the solutions for $y$ :
$y=0, \quad y=\frac{4}{25}$

STEP 7

Substitute $x^{2}$ back in for $y$ to find the solutions for $x$ :
$x^{2}=0, \quad x^{2}=\frac{4}{25}$

STEP 8

olving these equations gives the zeros of the function $x=0, \quad x=\pm\frac{2}{5}$ The zeros of the function are $x=0$ , $x=\frac{2}{5}$ , and $x=-\frac{2}{5}$ .

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QuestionFind the zeros of the function f(x)=−25x4+4x2f(x)=-25 x^{4}+4 x^{2}f(x)=−25x4+4x2. Enter answers as a comma-separated list.

Studdy Solution

STEP 1

Assumptions1. The function is given by f(x)=−25x4+4xf(x)=-25x^{4}+4x^{}f(x)=−25x4+4x . We need to find the zeros of the function, which are the values of xxx that make f(x)=0f(x)=0f(x)=0

STEP 2

To find the zeros of the function, we need to set the function equal to zero and solve for xxx.−25x4+4x2=0-25x^{4}+4x^{2}=0−25x4+4x2=0

STEP 3

We can simplify this equation by dividing all terms by -1.25x−x2=025x^{}-x^{2}=025x−x2=0

STEP 4

This is a quadratic equation in the form of ax4+bx2+c=0ax^{4}+bx^{2}+c=0ax4+bx2+c=0, where a=25a=25a=25, b=−4b=-4b=−4, and c=0c=0c=0. To solve this, we can use the substitution y=x2y=x^{2}y=x2 to turn it into a quadratic equation in yyy.25y2−4y=025y^{2}-4y=025y2−4y=0

STEP 5

This equation can be factored as followsy(25y−4)=0y(25y-4)=0y(25y−4)=0

STEP 6

Setting each factor equal to zero gives the solutions for yyy:y=0,y=425y=0, \quad y=\frac{4}{25}y=0,y=254​

STEP 7

Substitute x2x^{2}x2 back in for yyy to find the solutions for xxx:x2=0,x2=425x^{2}=0, \quad x^{2}=\frac{4}{25}x2=0,x2=254​

STEP 8

olving these equations gives the zeros of the functionx=0,x=±25x=0, \quad x=\pm\frac{2}{5}x=0,x=±52​The zeros of the function are x=0x=0x=0, x=25x=\frac{2}{5}x=52​, and x=−25x=-\frac{2}{5}x=−52​.

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QuestionFind the zeros of the function $f(x)=-25 x^{4}+4 x^{2}$ . Enter answers as a comma-separated list.

Assumptions1. The function is given by $f(x)=-25x^{4}+4x^{}$ . We need to find the zeros of the function, which are the values of $x$ that make $f(x)=0$

To find the zeros of the function, we need to set the function equal to zero and solve for $x$ .
$-25x^{4}+4x^{2}=0$

We can simplify this equation by dividing all terms by -1.
$25x^{}-x^{2}=0$

This is a quadratic equation in the form of $ax^{4}+bx^{2}+c=0$ , where $a=25$ , $b=-4$ , and $c=0$ . To solve this, we can use the substitution $y=x^{2}$ to turn it into a quadratic equation in $y$ .
$25y^{2}-4y=0$

This equation can be factored as follows $y(25y-4)=0$

Setting each factor equal to zero gives the solutions for $y$ :
$y=0, \quad y=\frac{4}{25}$

Substitute $x^{2}$ back in for $y$ to find the solutions for $x$ :
$x^{2}=0, \quad x^{2}=\frac{4}{25}$

olving these equations gives the zeros of the function $x=0, \quad x=\pm\frac{2}{5}$ The zeros of the function are $x=0$ , $x=\frac{2}{5}$ , and $x=-\frac{2}{5}$ .