Answer :
To find the roots of the function [tex]\( f(x) = x^4 + 21x^2 - 100 \)[/tex], let's break down the process:
1. Substitution: Notice that the function is a quartic equation, but can be simplified using substitution. Let [tex]\( y = x^2 \)[/tex]. Then the function becomes:
[tex]\[
f(y) = y^2 + 21y - 100
\][/tex]
2. Solve the Quadratic Equation: Now, we solve the quadratic equation for [tex]\( y \)[/tex]:
[tex]\[
y^2 + 21y - 100 = 0
\][/tex]
We can use the quadratic formula, [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 21 \)[/tex], and [tex]\( c = -100 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = 21^2 - 4 \times 1 \times (-100) = 441 + 400 = 841
\][/tex]
- Since the discriminant is positive, we have two real roots:
[tex]\[
y = \frac{-21 \pm \sqrt{841}}{2}
\][/tex]
- Calculate the roots:
[tex]\[
y = \frac{-21 + 29}{2} = 4
\][/tex]
[tex]\[
y = \frac{-21 - 29}{2} = -25
\][/tex]
3. Back-Substitution: Replace [tex]\( y \)[/tex] back with [tex]\( x^2 \)[/tex] to find the [tex]\( x \)[/tex] values.
- For [tex]\( y = 4 \)[/tex]:
[tex]\[
x^2 = 4 \implies x = \pm 2
\][/tex]
- For [tex]\( y = -25 \)[/tex]:
[tex]\[
x^2 = -25 \implies x = \pm 5i
\][/tex]
(since [tex]\( \sqrt{-25} = 5i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit)
4. Roots: Therefore, the roots of the original equation [tex]\( f(x) = x^4 + 21x^2 - 100 \)[/tex] are:
[tex]\[
x = -2, \, 2, \, -5i, \, 5i
\][/tex]
These steps help derive the roots of the function effectively, providing both real and imaginary solutions.
1. Substitution: Notice that the function is a quartic equation, but can be simplified using substitution. Let [tex]\( y = x^2 \)[/tex]. Then the function becomes:
[tex]\[
f(y) = y^2 + 21y - 100
\][/tex]
2. Solve the Quadratic Equation: Now, we solve the quadratic equation for [tex]\( y \)[/tex]:
[tex]\[
y^2 + 21y - 100 = 0
\][/tex]
We can use the quadratic formula, [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 21 \)[/tex], and [tex]\( c = -100 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = 21^2 - 4 \times 1 \times (-100) = 441 + 400 = 841
\][/tex]
- Since the discriminant is positive, we have two real roots:
[tex]\[
y = \frac{-21 \pm \sqrt{841}}{2}
\][/tex]
- Calculate the roots:
[tex]\[
y = \frac{-21 + 29}{2} = 4
\][/tex]
[tex]\[
y = \frac{-21 - 29}{2} = -25
\][/tex]
3. Back-Substitution: Replace [tex]\( y \)[/tex] back with [tex]\( x^2 \)[/tex] to find the [tex]\( x \)[/tex] values.
- For [tex]\( y = 4 \)[/tex]:
[tex]\[
x^2 = 4 \implies x = \pm 2
\][/tex]
- For [tex]\( y = -25 \)[/tex]:
[tex]\[
x^2 = -25 \implies x = \pm 5i
\][/tex]
(since [tex]\( \sqrt{-25} = 5i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit)
4. Roots: Therefore, the roots of the original equation [tex]\( f(x) = x^4 + 21x^2 - 100 \)[/tex] are:
[tex]\[
x = -2, \, 2, \, -5i, \, 5i
\][/tex]
These steps help derive the roots of the function effectively, providing both real and imaginary solutions.
