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Answer :
To solve the equation [tex]\(3x^4 = 27x^2\)[/tex] by factoring, follow these steps:
1. Rearrange the Equation:
Start by bringing all terms to one side of the equation:
[tex]\[
3x^4 - 27x^2 = 0
\][/tex]
2. Factor Out the Common Term:
Notice that both terms on the left-hand side have a common factor of [tex]\(3x^2\)[/tex]:
[tex]\[
3x^2(x^2 - 9) = 0
\][/tex]
3. Factor Further:
Recognize that [tex]\(x^2 - 9\)[/tex] is a difference of squares, which can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
So the equation becomes:
[tex]\[
3x^2(x - 3)(x + 3) = 0
\][/tex]
4. Apply the Zero Product Property:
For the product of factors to equal zero, at least one of the factors must be zero. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
- [tex]\(3x^2 = 0\)[/tex]
- Divide both sides by 3: [tex]\(x^2 = 0\)[/tex]
- Take the square root of both sides: [tex]\(x = 0\)[/tex]
- [tex]\(x - 3 = 0\)[/tex]
- Add 3 to both sides: [tex]\(x = 3\)[/tex]
- [tex]\(x + 3 = 0\)[/tex]
- Subtract 3 from both sides: [tex]\(x = -3\)[/tex]
5. Write the Solutions:
The solutions to the equation [tex]\(3x^4 = 27x^2\)[/tex] are [tex]\(x = 0\)[/tex], [tex]\(x = 3\)[/tex], and [tex]\(x = -3\)[/tex].
Thus, the equation has three solutions: [tex]\(x = 0\)[/tex], [tex]\(x = 3\)[/tex], and [tex]\(x = -3\)[/tex].
1. Rearrange the Equation:
Start by bringing all terms to one side of the equation:
[tex]\[
3x^4 - 27x^2 = 0
\][/tex]
2. Factor Out the Common Term:
Notice that both terms on the left-hand side have a common factor of [tex]\(3x^2\)[/tex]:
[tex]\[
3x^2(x^2 - 9) = 0
\][/tex]
3. Factor Further:
Recognize that [tex]\(x^2 - 9\)[/tex] is a difference of squares, which can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
So the equation becomes:
[tex]\[
3x^2(x - 3)(x + 3) = 0
\][/tex]
4. Apply the Zero Product Property:
For the product of factors to equal zero, at least one of the factors must be zero. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
- [tex]\(3x^2 = 0\)[/tex]
- Divide both sides by 3: [tex]\(x^2 = 0\)[/tex]
- Take the square root of both sides: [tex]\(x = 0\)[/tex]
- [tex]\(x - 3 = 0\)[/tex]
- Add 3 to both sides: [tex]\(x = 3\)[/tex]
- [tex]\(x + 3 = 0\)[/tex]
- Subtract 3 from both sides: [tex]\(x = -3\)[/tex]
5. Write the Solutions:
The solutions to the equation [tex]\(3x^4 = 27x^2\)[/tex] are [tex]\(x = 0\)[/tex], [tex]\(x = 3\)[/tex], and [tex]\(x = -3\)[/tex].
Thus, the equation has three solutions: [tex]\(x = 0\)[/tex], [tex]\(x = 3\)[/tex], and [tex]\(x = -3\)[/tex].
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