Answer :

To solve the equation [tex]\(2x^4 = 9x^2\)[/tex], follow these steps:

1. Rearrange the Equation:
Start by bringing all the terms to one side of the equation to set it to zero. This helps in solving the equation:
[tex]\[
2x^4 - 9x^2 = 0
\][/tex]

2. Factor the Equation:
Notice that there is a common factor of [tex]\(x^2\)[/tex] in both terms on the left:
[tex]\[
x^2(2x^2 - 9) = 0
\][/tex]

3. Apply the Zero Product Property:
The zero product property states that if a product of factors equals zero, at least one of the factors must be zero. Thus, we set each factor equal to zero:

a) [tex]\(x^2 = 0\)[/tex]

b) [tex]\(2x^2 - 9 = 0\)[/tex]

4. Solve Each Factor:

a) For [tex]\(x^2 = 0\)[/tex]:
- Take the square root of both sides:
[tex]\[
x = 0
\][/tex]

b) For [tex]\(2x^2 - 9 = 0\)[/tex]:
- Add 9 to both sides:
[tex]\[
2x^2 = 9
\][/tex]
- Divide both sides by 2:
[tex]\[
x^2 = \frac{9}{2}
\][/tex]
- Take the square root of both sides. Remember to consider both the positive and negative roots:
[tex]\[
x = \pm \sqrt{\frac{9}{2}} = \pm \frac{3\sqrt{2}}{2}
\][/tex]

5. List the Solutions:
Therefore, the solutions to the equation [tex]\(2x^4 = 9x^2\)[/tex] are:
[tex]\[
x = 0, \quad x = -\frac{3\sqrt{2}}{2}, \quad x = \frac{3\sqrt{2}}{2}
\][/tex]

These are the complete set of solutions for the given equation.

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Rewritten by : Barada