IDNLearn.com offers a comprehensive solution for all your question and answer needs. Ask your questions and receive comprehensive, trustworthy responses from our dedicated team of experts.

The following equation has three real solutions, one of which is zero:

22x^4 = 3x^2

Find the other (non-zero) solutions. List your answers separated by a comma. For example, if you found the solutions x=1 and x=2, you would enter 1,2.


Sagot :

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]