To solve the equation [tex]\( 22x^4 = 3x^2 \)[/tex], follow these steps:
1. Rewrite the equation:
Start by arranging the equation in standard polynomial form. Move all terms to one side of the equation:
[tex]\[
22x^4 - 3x^2 = 0
\][/tex]
2. Factor the equation:
Factor out the common term [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(22x^2 - 3) = 0
\][/tex]
3. Set each factor equal to zero and solve:
The equation [tex]\( x^2(22x^2 - 3) = 0 \)[/tex] implies two possible cases, based on the zero product property:
Case 1:
[tex]\[
x^2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 0
\][/tex]
This is one of the solutions.
Case 2:
[tex]\[
22x^2 - 3 = 0
\][/tex]
Solving for [tex]\( x^2 \)[/tex] by isolating the term:
[tex]\[
22x^2 = 3
\][/tex]
Divide by 22:
[tex]\[
x^2 = \frac{3}{22}
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm \sqrt{\frac{3}{22}}
\][/tex]
4. Simplify the square root expression:
We can leave the solution in its exact form or approximate the square root numerically:
[tex]\[
x = \pm \sqrt{\frac{3}{22}} \approx \pm 0.369
\][/tex]
So, the other (non-zero) solutions to the equation are:
[tex]\[
x \approx -0.369, 0.369
\][/tex]
Hence, the non-zero solutions to the equation [tex]\( 22x^4 = 3x^2 \)[/tex] are:
[tex]\[
-0.369274472937998, 0.369274472937998
\][/tex]
