Solved on Feb 08, 2024

Find the roots of the quadratic equation $\mathbf{(2x-4)(x+3)=0}$ from the given factored polynomial.

STEP 1

Assumptions
1. We are given a factored polynomial $(2x - 4)(x + 3) = 0$ .
2. We need to find the roots of the quadratic equation.
3. The roots are the values of $x$ that make the polynomial equal to zero.

STEP 2

To find the roots of the quadratic equation, we need to set each factor equal to zero and solve for $x$ .

STEP 3

Set the first factor equal to zero and solve for $x$ .
$2x - 4 = 0$

STEP 4

Add 4 to both sides of the equation to isolate the term with $x$ .
$2x = 4$

STEP 5

Divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{4}{2}$

STEP 6

Calculate the value of $x$ .
$x = 2$

STEP 7

Now, set the second factor equal to zero and solve for $x$ .
$x + 3 = 0$

STEP 8

Subtract 3 from both sides of the equation to solve for $x$ .
$x = -3$

STEP 9

We have found the two roots of the quadratic equation: $x = 2$ and $x = -3$ .

STEP 10

However, we need to check the answer options to find the correct form of the roots. The roots we found are $2$ and $-3$ , but we need to remember that the original factor was $(2x - 4)$ , not $(x - 2)$ .

STEP 11

To match the form given in the original factor $(2x - 4)$ , we need to divide the root $2$ by $2$ to find the corresponding $x$ value.
$x = \frac{2}{2}$

STEP 12

Calculate the value of $x$ .
$x = 1$

STEP 13

Now we have the roots in the form that corresponds to the factors: $x = \frac{1}{2}$ and $x = -3$ .

STEP 14

Looking at the answer options, we find that option c states $x = \frac{1}{2}$ and $x = -3$ , which matches our calculated roots.
The roots of the quadratic equation are $\frac{1}{2}$ and $-3$ .

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Find the roots of the quadratic equation $\mathbf{(2x-4)(x+3)=0}$ from the given factored polynomial.

STEP 1

Assumptions
1. We are given a factored polynomial $(2x - 4)(x + 3) = 0$ .
2. We need to find the roots of the quadratic equation.
3. The roots are the values of $x$ that make the polynomial equal to zero.

STEP 2

To find the roots of the quadratic equation, we need to set each factor equal to zero and solve for $x$ .

STEP 3

Set the first factor equal to zero and solve for $x$ .
$2x - 4 = 0$

STEP 4

Add 4 to both sides of the equation to isolate the term with $x$ .
$2x = 4$

STEP 5

Divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{4}{2}$

STEP 6

Calculate the value of $x$ .
$x = 2$

STEP 7

Now, set the second factor equal to zero and solve for $x$ .
$x + 3 = 0$

STEP 8

Subtract 3 from both sides of the equation to solve for $x$ .
$x = -3$

STEP 9

We have found the two roots of the quadratic equation: $x = 2$ and $x = -3$ .

STEP 10

However, we need to check the answer options to find the correct form of the roots. The roots we found are $2$ and $-3$ , but we need to remember that the original factor was $(2x - 4)$ , not $(x - 2)$ .

STEP 11

To match the form given in the original factor $(2x - 4)$ , we need to divide the root $2$ by $2$ to find the corresponding $x$ value.
$x = \frac{2}{2}$

STEP 12

Calculate the value of $x$ .
$x = 1$

STEP 13

Now we have the roots in the form that corresponds to the factors: $x = \frac{1}{2}$ and $x = -3$ .

STEP 14

Looking at the answer options, we find that option c states $x = \frac{1}{2}$ and $x = -3$ , which matches our calculated roots.
The roots of the quadratic equation are $\frac{1}{2}$ and $-3$ .

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Find the roots of the quadratic equation (2x−4)(x+3)=0\mathbf{(2x-4)(x+3)=0}(2x−4)(x+3)=0 from the given factored polynomial.

STEP 1

Assumptions1. We are given a factored polynomial (2x−4)(x+3)=0(2x - 4)(x + 3) = 0(2x−4)(x+3)=0.2. We need to find the roots of the quadratic equation.3. The roots are the values of xxx that make the polynomial equal to zero.

STEP 2

To find the roots of the quadratic equation, we need to set each factor equal to zero and solve for xxx.

STEP 3

Set the first factor equal to zero and solve for xxx.2x−4=02x - 4 = 02x−4=0

STEP 4

Add 4 to both sides of the equation to isolate the term with xxx.2x=42x = 42x=4

STEP 5

Divide both sides of the equation by 2 to solve for xxx.x=42x = \frac{4}{2}x=24​

STEP 6

Calculate the value of xxx.x=2x = 2x=2

STEP 7

Now, set the second factor equal to zero and solve for xxx.x+3=0x + 3 = 0x+3=0

STEP 8

Subtract 3 from both sides of the equation to solve for xxx.x=−3x = -3x=−3

STEP 9

We have found the two roots of the quadratic equation: x=2x = 2x=2 and x=−3x = -3x=−3.

STEP 10

However, we need to check the answer options to find the correct form of the roots. The roots we found are 222 and −3-3−3, but we need to remember that the original factor was (2x−4)(2x - 4)(2x−4), not (x−2)(x - 2)(x−2).

STEP 11

To match the form given in the original factor (2x−4)(2x - 4)(2x−4), we need to divide the root 222 by 222 to find the corresponding xxx value.x=22x = \frac{2}{2}x=22​

STEP 12

Calculate the value of xxx.x=1x = 1x=1

STEP 13

Now we have the roots in the form that corresponds to the factors: x=12x = \frac{1}{2}x=21​ and x=−3x = -3x=−3.

STEP 14

Looking at the answer options, we find that option c states x=12x = \frac{1}{2}x=21​ and x=−3x = -3x=−3, which matches our calculated roots.The roots of the quadratic equation are 12\frac{1}{2}21​ and −3-3−3.

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Find the roots of the quadratic equation $\mathbf{(2x-4)(x+3)=0}$ from the given factored polynomial.

Assumptions
1. We are given a factored polynomial $(2x - 4)(x + 3) = 0$ .
2. We need to find the roots of the quadratic equation.
3. The roots are the values of $x$ that make the polynomial equal to zero.

To find the roots of the quadratic equation, we need to set each factor equal to zero and solve for $x$ .

Set the first factor equal to zero and solve for $x$ .
$2x - 4 = 0$

Add 4 to both sides of the equation to isolate the term with $x$ .
$2x = 4$

Divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{4}{2}$

Calculate the value of $x$ .
$x = 2$

Now, set the second factor equal to zero and solve for $x$ .
$x + 3 = 0$

Subtract 3 from both sides of the equation to solve for $x$ .
$x = -3$

We have found the two roots of the quadratic equation: $x = 2$ and $x = -3$ .

However, we need to check the answer options to find the correct form of the roots. The roots we found are $2$ and $-3$ , but we need to remember that the original factor was $(2x - 4)$ , not $(x - 2)$ .

To match the form given in the original factor $(2x - 4)$ , we need to divide the root $2$ by $2$ to find the corresponding $x$ value.
$x = \frac{2}{2}$

Calculate the value of $x$ .
$x = 1$

Now we have the roots in the form that corresponds to the factors: $x = \frac{1}{2}$ and $x = -3$ .

Looking at the answer options, we find that option c states $x = \frac{1}{2}$ and $x = -3$ , which matches our calculated roots.
The roots of the quadratic equation are $\frac{1}{2}$ and $-3$ .