Answer :
To solve the equation [tex]\(2x^4 = 9x^2\)[/tex], we will follow these steps:
1. Set up the equation: Start with the equation [tex]\(2x^4 = 9x^2\)[/tex].
2. Move all terms to one side: Subtract [tex]\(9x^2\)[/tex] from both sides to get:
[tex]\[
2x^4 - 9x^2 = 0
\][/tex]
3. Factor the equation: Notice that there is a common factor of [tex]\(x^2\)[/tex]. Factor [tex]\(x^2\)[/tex] out:
[tex]\[
x^2(2x^2 - 9) = 0
\][/tex]
4. Solve each factor separately:
- First factor: [tex]\(x^2 = 0\)[/tex]
- Solve for [tex]\(x\)[/tex]: [tex]\(x = 0\)[/tex]
- Second factor: [tex]\(2x^2 - 9 = 0\)[/tex]
- Add [tex]\(9\)[/tex] to both sides:
[tex]\[
2x^2 = 9
\][/tex]
- Divide both sides by [tex]\(2\)[/tex]:
[tex]\[
x^2 = \frac{9}{2}
\][/tex]
- Take the square root of both sides:
[tex]\[
x = \pm \sqrt{\frac{9}{2}}
\][/tex]
5. Simplify the square roots:
- [tex]\(\sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}}\)[/tex]
- Multiply numerator and denominator by [tex]\(\sqrt{2}\)[/tex] to rationalize the denominator:
[tex]\[
x = \pm \frac{3\sqrt{2}}{2}
\][/tex]
6. Summarize the solutions: Therefore, the solutions to the equation [tex]\(2x^4 = 9x^2\)[/tex] are:
- [tex]\(x = 0\)[/tex]
- [tex]\(x = -\frac{3\sqrt{2}}{2}\)[/tex]
- [tex]\(x = \frac{3\sqrt{2}}{2}\)[/tex]
These are the values for [tex]\(x\)[/tex] that satisfy the given equation.
1. Set up the equation: Start with the equation [tex]\(2x^4 = 9x^2\)[/tex].
2. Move all terms to one side: Subtract [tex]\(9x^2\)[/tex] from both sides to get:
[tex]\[
2x^4 - 9x^2 = 0
\][/tex]
3. Factor the equation: Notice that there is a common factor of [tex]\(x^2\)[/tex]. Factor [tex]\(x^2\)[/tex] out:
[tex]\[
x^2(2x^2 - 9) = 0
\][/tex]
4. Solve each factor separately:
- First factor: [tex]\(x^2 = 0\)[/tex]
- Solve for [tex]\(x\)[/tex]: [tex]\(x = 0\)[/tex]
- Second factor: [tex]\(2x^2 - 9 = 0\)[/tex]
- Add [tex]\(9\)[/tex] to both sides:
[tex]\[
2x^2 = 9
\][/tex]
- Divide both sides by [tex]\(2\)[/tex]:
[tex]\[
x^2 = \frac{9}{2}
\][/tex]
- Take the square root of both sides:
[tex]\[
x = \pm \sqrt{\frac{9}{2}}
\][/tex]
5. Simplify the square roots:
- [tex]\(\sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}}\)[/tex]
- Multiply numerator and denominator by [tex]\(\sqrt{2}\)[/tex] to rationalize the denominator:
[tex]\[
x = \pm \frac{3\sqrt{2}}{2}
\][/tex]
6. Summarize the solutions: Therefore, the solutions to the equation [tex]\(2x^4 = 9x^2\)[/tex] are:
- [tex]\(x = 0\)[/tex]
- [tex]\(x = -\frac{3\sqrt{2}}{2}\)[/tex]
- [tex]\(x = \frac{3\sqrt{2}}{2}\)[/tex]
These are the values for [tex]\(x\)[/tex] that satisfy the given equation.
