Answer :

To solve the equation [tex]\(7x^4 - 14x^3 + 21x^2 = 0\)[/tex], we can follow these steps:

1. Factor out the greatest common factor:
All terms in the polynomial have a factor of [tex]\(7x^2\)[/tex]. So, we factor [tex]\(7x^2\)[/tex] out of the expression:
[tex]\[
7x^2(x^2 - 2x + 3) = 0
\][/tex]

2. Apply the zero-product property:
According to the zero-product property, if a product of factors equals zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve:

a. First factor [tex]\(7x^2 = 0\)[/tex]:
[tex]\[
7x^2 = 0 \quad \Rightarrow \quad x^2 = 0 \quad \Rightarrow \quad x = 0
\][/tex]

b. Second factor [tex]\(x^2 - 2x + 3 = 0\)[/tex]:
This is a quadratic equation, which we can solve using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 3\)[/tex].

Plugging in these values, we get:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}
\][/tex]
Simplifying inside the square root:
[tex]\[
x = \frac{2 \pm \sqrt{4 - 12}}{2} = \frac{2 \pm \sqrt{-8}}{2}
\][/tex]

Since [tex]\(\sqrt{-8}\)[/tex] involves an imaginary number, we express this as:
[tex]\[
\sqrt{-8} = \sqrt{-1 \cdot 4 \cdot 2} = \sqrt{-1} \cdot \sqrt{4} \cdot \sqrt{2} = 2i\sqrt{2}
\][/tex]

Therefore, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
x = \frac{2 \pm 2i\sqrt{2}}{2}
\][/tex]
Simplifying:
[tex]\[
x = 1 \pm i\sqrt{2}
\][/tex]

3. List the solutions:
The solutions to the equation [tex]\(7x^4 - 14x^3 + 21x^2 = 0\)[/tex] are:
- [tex]\(x = 0\)[/tex]
- [tex]\(x = 1 - i\sqrt{2}\)[/tex]
- [tex]\(x = 1 + i\sqrt{2}\)[/tex]

These are the values of [tex]\(x\)[/tex] that satisfy the equation.

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