Answer :
To solve the equation [tex]\(x^4 + 4x^2 - 21 = 0\)[/tex], we can start by using a substitution method to make it easier. Let's proceed step-by-step:
1. Substitution: Let [tex]\( y = x^2 \)[/tex]. This means [tex]\( x^4 = y^2 \)[/tex]. Substitute these into the original equation to get:
[tex]\[
y^2 + 4y - 21 = 0
\][/tex]
2. Solve the Quadratic Equation: The equation [tex]\( y^2 + 4y - 21 = 0 \)[/tex] is a standard quadratic equation. We can solve it using the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -21\)[/tex].
3. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 4^2 - 4 \times 1 \times (-21) = 16 + 84 = 100
\][/tex]
4. Find the Roots for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-4 \pm \sqrt{100}}{2 \times 1} = \frac{-4 \pm 10}{2}
\][/tex]
This gives us two solutions for [tex]\(y\)[/tex]:
- [tex]\( y = \frac{6}{2} = 3 \)[/tex]
- [tex]\( y = \frac{-14}{2} = -7 \)[/tex]
5. Back-Substitution to Find [tex]\(x\)[/tex]:
- For [tex]\( y = 3 \)[/tex], we have [tex]\( x^2 = 3 \)[/tex]. This gives:
[tex]\[
x = \pm \sqrt{3}
\][/tex]
- For [tex]\( y = -7 \)[/tex], we have [tex]\( x^2 = -7 \)[/tex]. This means:
[tex]\[
x = \pm \sqrt{-7} = \pm \sqrt{7}i
\][/tex]
(since [tex]\(\sqrt{-1} = i\)[/tex], the imaginary unit)
6. Solution: Collecting all these values, the solutions for the initial equation [tex]\(x^4 + 4x^2 - 21 = 0\)[/tex] are:
- [tex]\(x = -\sqrt{3}\)[/tex]
- [tex]\(x = \sqrt{3}\)[/tex]
- [tex]\(x = -\sqrt{7}i\)[/tex]
- [tex]\(x = \sqrt{7}i\)[/tex]
Therefore, the equation [tex]\(x^4 + 4x^2 - 21 = 0\)[/tex] has the solutions [tex]\( -\sqrt{3}, \sqrt{3}, -\sqrt{7}i, \sqrt{7}i \)[/tex].
1. Substitution: Let [tex]\( y = x^2 \)[/tex]. This means [tex]\( x^4 = y^2 \)[/tex]. Substitute these into the original equation to get:
[tex]\[
y^2 + 4y - 21 = 0
\][/tex]
2. Solve the Quadratic Equation: The equation [tex]\( y^2 + 4y - 21 = 0 \)[/tex] is a standard quadratic equation. We can solve it using the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -21\)[/tex].
3. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 4^2 - 4 \times 1 \times (-21) = 16 + 84 = 100
\][/tex]
4. Find the Roots for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-4 \pm \sqrt{100}}{2 \times 1} = \frac{-4 \pm 10}{2}
\][/tex]
This gives us two solutions for [tex]\(y\)[/tex]:
- [tex]\( y = \frac{6}{2} = 3 \)[/tex]
- [tex]\( y = \frac{-14}{2} = -7 \)[/tex]
5. Back-Substitution to Find [tex]\(x\)[/tex]:
- For [tex]\( y = 3 \)[/tex], we have [tex]\( x^2 = 3 \)[/tex]. This gives:
[tex]\[
x = \pm \sqrt{3}
\][/tex]
- For [tex]\( y = -7 \)[/tex], we have [tex]\( x^2 = -7 \)[/tex]. This means:
[tex]\[
x = \pm \sqrt{-7} = \pm \sqrt{7}i
\][/tex]
(since [tex]\(\sqrt{-1} = i\)[/tex], the imaginary unit)
6. Solution: Collecting all these values, the solutions for the initial equation [tex]\(x^4 + 4x^2 - 21 = 0\)[/tex] are:
- [tex]\(x = -\sqrt{3}\)[/tex]
- [tex]\(x = \sqrt{3}\)[/tex]
- [tex]\(x = -\sqrt{7}i\)[/tex]
- [tex]\(x = \sqrt{7}i\)[/tex]
Therefore, the equation [tex]\(x^4 + 4x^2 - 21 = 0\)[/tex] has the solutions [tex]\( -\sqrt{3}, \sqrt{3}, -\sqrt{7}i, \sqrt{7}i \)[/tex].
