Answer :
To find the real zeros of the function [tex]\( r(x) = x^4 - 21x^2 - 100 \)[/tex], we'll first rearrange and simplify the expression if possible.
1. Substitution: Notice that the function can be rewritten in terms of [tex]\( y \)[/tex] by substituting [tex]\( y = x^2 \)[/tex]. Then, the function becomes:
[tex]\[
r(y) = y^2 - 21y - 100
\][/tex]
This is a quadratic equation in terms of [tex]\( y \)[/tex].
2. Solve the Quadratic Equation: We'll use the quadratic formula to solve for [tex]\( y \)[/tex]. The quadratic formula is given by:
[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].
3. Calculate the Discriminant: First, calculate the discriminant [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
b^2 - 4ac = (-21)^2 - 4 \times 1 \times (-100) = 441 + 400 = 841
\][/tex]
4. Find the values of [tex]\( y \)[/tex]: Substitute back into the quadratic formula:
[tex]\[
y = \frac{21 \pm \sqrt{841}}{2}
\][/tex]
Since [tex]\( \sqrt{841} = 29 \)[/tex], the equation becomes:
[tex]\[
y = \frac{21 \pm 29}{2}
\][/tex]
This gives us two possible solutions for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{21 + 29}{2} = 25 \quad \text{and} \quad y = \frac{21 - 29}{2} = -4
\][/tex]
5. Solve for [tex]\( x \)[/tex]: Remember that [tex]\( y = x^2 \)[/tex], so we solve for [tex]\( x \)[/tex] in each case:
- For [tex]\( y = 25 \)[/tex], [tex]\( x^2 = 25 \)[/tex] gives us [tex]\( x = \pm 5 \)[/tex].
- For [tex]\( y = -4 \)[/tex], [tex]\( x^2 = -4 \)[/tex] gives us complex solutions [tex]\( x = \pm 2i \)[/tex], which are not real.
Therefore, the real zeros of the function [tex]\( r(x) = x^4 - 21x^2 - 100 \)[/tex] are [tex]\( x = -5 \)[/tex] and [tex]\( x = 5 \)[/tex].
1. Substitution: Notice that the function can be rewritten in terms of [tex]\( y \)[/tex] by substituting [tex]\( y = x^2 \)[/tex]. Then, the function becomes:
[tex]\[
r(y) = y^2 - 21y - 100
\][/tex]
This is a quadratic equation in terms of [tex]\( y \)[/tex].
2. Solve the Quadratic Equation: We'll use the quadratic formula to solve for [tex]\( y \)[/tex]. The quadratic formula is given by:
[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].
3. Calculate the Discriminant: First, calculate the discriminant [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
b^2 - 4ac = (-21)^2 - 4 \times 1 \times (-100) = 441 + 400 = 841
\][/tex]
4. Find the values of [tex]\( y \)[/tex]: Substitute back into the quadratic formula:
[tex]\[
y = \frac{21 \pm \sqrt{841}}{2}
\][/tex]
Since [tex]\( \sqrt{841} = 29 \)[/tex], the equation becomes:
[tex]\[
y = \frac{21 \pm 29}{2}
\][/tex]
This gives us two possible solutions for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{21 + 29}{2} = 25 \quad \text{and} \quad y = \frac{21 - 29}{2} = -4
\][/tex]
5. Solve for [tex]\( x \)[/tex]: Remember that [tex]\( y = x^2 \)[/tex], so we solve for [tex]\( x \)[/tex] in each case:
- For [tex]\( y = 25 \)[/tex], [tex]\( x^2 = 25 \)[/tex] gives us [tex]\( x = \pm 5 \)[/tex].
- For [tex]\( y = -4 \)[/tex], [tex]\( x^2 = -4 \)[/tex] gives us complex solutions [tex]\( x = \pm 2i \)[/tex], which are not real.
Therefore, the real zeros of the function [tex]\( r(x) = x^4 - 21x^2 - 100 \)[/tex] are [tex]\( x = -5 \)[/tex] and [tex]\( x = 5 \)[/tex].
