QuestionSolve the equation graphically: $3 x^{4}+6 x^{2}+1=0$ with the solution $x=1$ .

Studdy Solution

STEP 1

Assumptions1. The equation to solve is $3x^4 +6x^ +1 =0$ . . We are given that $x =1$ is a solution.
3. We will use the graphical method to solve the equation.

STEP 2

First, we need to rewrite the equation in a form that can be graphed. The equation is already in a form that can be graphed, so we can proceed to the next step.

STEP 3

Now, we will graph the function $y =3x^ +6x^2 +1$ .

STEP 4

The graph of the function will be a curve that opens upwards because the leading coefficient (3) is positive and the degree of the polynomial is even (4).

STEP 5

We are given that $x =1$ is a solution to the equation. This means that when $x =1$ , the function $y =3x^4 +x^2 +1$ equals zero. We can check this by substituting $x =1$ into the equation.

STEP 6

Substitute $x =1$ into the equation.
$y =3(1)^4 +6(1)^2 +1$

STEP 7

Calculate the value of $y$ when $x =1$ .
$y =3(1) +6(1) +1 =10$

STEP 8

The value of $y$ when $x =1$ is not zero, so $x =1$ is not a solution to the equation $3x^4 +6x^2 +1 =0$ .

STEP 9

To find the solutions to the equation, we need to find the $x$ values where the graph of the function intersects the x-axis. This is because the solutions to the equation are the $x$ values where $y =$ .

STEP 10

The graph of the function $y =3x^4 +6x^2 +$ does not intersect the x-axis, so the equation $3x^4 +6x^2 + =0$ has no real solutions.

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QuestionSolve the equation graphically: $3 x^{4}+6 x^{2}+1=0$ with the solution $x=1$ .

Studdy Solution

STEP 1

Assumptions1. The equation to solve is $3x^4 +6x^ +1 =0$ . . We are given that $x =1$ is a solution.
3. We will use the graphical method to solve the equation.

STEP 2

First, we need to rewrite the equation in a form that can be graphed. The equation is already in a form that can be graphed, so we can proceed to the next step.

STEP 3

Now, we will graph the function $y =3x^ +6x^2 +1$ .

STEP 4

The graph of the function will be a curve that opens upwards because the leading coefficient (3) is positive and the degree of the polynomial is even (4).

STEP 5

We are given that $x =1$ is a solution to the equation. This means that when $x =1$ , the function $y =3x^4 +x^2 +1$ equals zero. We can check this by substituting $x =1$ into the equation.

STEP 6

Substitute $x =1$ into the equation.
$y =3(1)^4 +6(1)^2 +1$

STEP 7

Calculate the value of $y$ when $x =1$ .
$y =3(1) +6(1) +1 =10$

STEP 8

The value of $y$ when $x =1$ is not zero, so $x =1$ is not a solution to the equation $3x^4 +6x^2 +1 =0$ .

STEP 9

To find the solutions to the equation, we need to find the $x$ values where the graph of the function intersects the x-axis. This is because the solutions to the equation are the $x$ values where $y =$ .

STEP 10

The graph of the function $y =3x^4 +6x^2 +$ does not intersect the x-axis, so the equation $3x^4 +6x^2 + =0$ has no real solutions.

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QuestionSolve the equation graphically: 3x4+6x2+1=03 x^{4}+6 x^{2}+1=03x4+6x2+1=0 with the solution x=1x=1x=1.

Studdy Solution

STEP 1

Assumptions1. The equation to solve is 3x4+6x+1=03x^4 +6x^ +1 =03x4+6x+1=0. . We are given that x=1x =1x=1 is a solution.3. We will use the graphical method to solve the equation.

STEP 2

First, we need to rewrite the equation in a form that can be graphed. The equation is already in a form that can be graphed, so we can proceed to the next step.

STEP 3

Now, we will graph the function y=3x+6x2+1y =3x^ +6x^2 +1y=3x+6x2+1.

STEP 4

The graph of the function will be a curve that opens upwards because the leading coefficient (3) is positive and the degree of the polynomial is even (4).

STEP 5

We are given that x=1x =1x=1 is a solution to the equation. This means that when x=1x =1x=1, the function y=3x4+x2+1y =3x^4 +x^2 +1y=3x4+x2+1 equals zero. We can check this by substituting x=1x =1x=1 into the equation.

STEP 6

Substitute x=1x =1x=1 into the equation.y=3(1)4+6(1)2+1y =3(1)^4 +6(1)^2 +1y=3(1)4+6(1)2+1

STEP 7

Calculate the value of yyy when x=1x =1x=1.y=3(1)+6(1)+1=10y =3(1) +6(1) +1 =10y=3(1)+6(1)+1=10

STEP 8

The value of yyy when x=1x =1x=1 is not zero, so x=1x =1x=1 is not a solution to the equation 3x4+6x2+1=03x^4 +6x^2 +1 =03x4+6x2+1=0.

STEP 9

To find the solutions to the equation, we need to find the xxx values where the graph of the function intersects the x-axis. This is because the solutions to the equation are the xxx values where y=y =y=.

STEP 10

The graph of the function y=3x4+6x2+y =3x^4 +6x^2 +y=3x4+6x2+ does not intersect the x-axis, so the equation 3x4+6x2+=03x^4 +6x^2 + =03x4+6x2+=0 has no real solutions.

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QuestionSolve the equation graphically: $3 x^{4}+6 x^{2}+1=0$ with the solution $x=1$ .

Assumptions1. The equation to solve is $3x^4 +6x^ +1 =0$ . . We are given that $x =1$ is a solution.
3. We will use the graphical method to solve the equation.

Now, we will graph the function $y =3x^ +6x^2 +1$ .

We are given that $x =1$ is a solution to the equation. This means that when $x =1$ , the function $y =3x^4 +x^2 +1$ equals zero. We can check this by substituting $x =1$ into the equation.

Substitute $x =1$ into the equation.
$y =3(1)^4 +6(1)^2 +1$

Calculate the value of $y$ when $x =1$ .
$y =3(1) +6(1) +1 =10$

The value of $y$ when $x =1$ is not zero, so $x =1$ is not a solution to the equation $3x^4 +6x^2 +1 =0$ .

To find the solutions to the equation, we need to find the $x$ values where the graph of the function intersects the x-axis. This is because the solutions to the equation are the $x$ values where $y =$ .

The graph of the function $y =3x^4 +6x^2 +$ does not intersect the x-axis, so the equation $3x^4 +6x^2 + =0$ has no real solutions.