To solve the equation [tex]\(x^4 - 10x^2 + 4 = 0\)[/tex], let's follow a step-by-step approach:
1. Substitute [tex]\( y = x^2 \)[/tex]: This substitution helps convert the equation from quartic to quadratic.
[tex]\[
x^4 - 10x^2 + 4 = 0 \implies y^2 - 10y + 4 = 0
\][/tex]
2. Solve the quadratic equation [tex]\( y^2 - 10y + 4 = 0 \)[/tex]:
Here, we can use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. In this case, [tex]\(a = 1\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = 4\)[/tex]. Plugging in these values:
[tex]\[
y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}
\][/tex]
Simplify under the square root:
[tex]\[
y = \frac{10 \pm \sqrt{100 - 16}}{2}
\][/tex]
[tex]\[
y = \frac{10 \pm \sqrt{84}}{2}
\][/tex]
Since [tex]\(\sqrt{84}\)[/tex] can be further simplified:
[tex]\[
y = \frac{10 \pm 2\sqrt{21}}{2}
\][/tex]
Divide by 2:
[tex]\[
y = 5 \pm \sqrt{21}
\][/tex]
Hence, we have two solutions for [tex]\( y \)[/tex]:
[tex]\[
y_1 = 5 + \sqrt{21} \quad \text{and} \quad y_2 = 5 - \sqrt{21}
\][/tex]
3. Re-substitute [tex]\( y = x^2 \)[/tex]: Now we will solve for [tex]\( x \)[/tex] in terms of these results.
For [tex]\( y = 5 + \sqrt{21} \)[/tex]:
[tex]\[
x^2 = 5 + \sqrt{21}
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm \sqrt{5 + \sqrt{21}}
\][/tex]
So, the solutions here are:
[tex]\[
x_1 = \sqrt{5 + \sqrt{21}} \quad \text{and} \quad x_2 = -\sqrt{5 + \sqrt{21}}
\][/tex]
For [tex]\( y = 5 - \sqrt{21} \)[/tex]:
[tex]\[
x^2 = 5 - \sqrt{21}
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm \sqrt{5 - \sqrt{21}}
\][/tex]
So, the solutions here are:
[tex]\[
x_3 = \sqrt{5 - \sqrt{21}} \quad \text{and} \quad x_4 = -\sqrt{5 - \sqrt{21}}
\][/tex]
4. List all solutions: Collect all the solutions found from both cases.
The roots of the equation [tex]\(x^4 - 10x^2 + 4 = 0\)[/tex] are:
[tex]\[
x = \sqrt{5 + \sqrt{21}}, \quad x = -\sqrt{5 + \sqrt{21}}, \quad x = \sqrt{5 - \sqrt{21}}, \quad \text{and} \quad x = -\sqrt{5 - \sqrt{21}}
\][/tex]
Thus, the complete solution set is:
[tex]\[
x = \left\{-\sqrt{5 - \sqrt{21}}, \, \sqrt{5 - \sqrt{21}}, \, -\sqrt{5 + \sqrt{21}}, \, \sqrt{5 + \sqrt{21}}\right\}
\][/tex]
