Answer :
To solve the polynomial equation [tex]\(4x^4 - 11x^2 - 45 = 0\)[/tex], follow these steps:
1. Substitution:
Start by letting [tex]\(y = x^2\)[/tex]. This transforms the original equation into a quadratic equation in terms of [tex]\(y\)[/tex]:
[tex]\[
4y^2 - 11y - 45 = 0
\][/tex]
2. Factor or Use the Quadratic Formula:
We will solve this quadratic equation. For easier numbers, often factoring could help, but in case it's challenging, the quadratic formula can be used:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = -11\)[/tex], and [tex]\(c = -45\)[/tex].
3. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = (-11)^2 - 4 \cdot 4 \cdot (-45) = 121 + 720 = 841
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{11 \pm \sqrt{841}}{8} = \frac{11 \pm 29}{8}
\][/tex]
This gives us two possible values for [tex]\(y\)[/tex]:
- [tex]\(y = \frac{11 + 29}{8} = \frac{40}{8} = 5\)[/tex]
- [tex]\(y = \frac{11 - 29}{8} = \frac{-18}{8} = -\frac{9}{4}\)[/tex]
5. Back-Substitute for [tex]\(x\)[/tex]:
Remember [tex]\(y = x^2\)[/tex], so substitute back:
- For [tex]\(y = 5\)[/tex], we have [tex]\(x^2 = 5\)[/tex]. This gives:
- [tex]\(x = \pm \sqrt{5}\)[/tex]
- For [tex]\(y = -\frac{9}{4}\)[/tex], we have [tex]\(x^2 = -\frac{9}{4}\)[/tex]. Since this is negative, the solutions will be complex:
- [tex]\(x = \pm \sqrt{-\frac{9}{4}} = \pm \frac{3}{2}i\)[/tex]
6. Final Solutions:
The complete solution set for the original polynomial equation is:
[tex]\[
x = -\sqrt{5}, \sqrt{5}, -\frac{3}{2}i, \frac{3}{2}i
\][/tex]
These are the four roots of the equation, including two real and two imaginary solutions.
1. Substitution:
Start by letting [tex]\(y = x^2\)[/tex]. This transforms the original equation into a quadratic equation in terms of [tex]\(y\)[/tex]:
[tex]\[
4y^2 - 11y - 45 = 0
\][/tex]
2. Factor or Use the Quadratic Formula:
We will solve this quadratic equation. For easier numbers, often factoring could help, but in case it's challenging, the quadratic formula can be used:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = -11\)[/tex], and [tex]\(c = -45\)[/tex].
3. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = (-11)^2 - 4 \cdot 4 \cdot (-45) = 121 + 720 = 841
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{11 \pm \sqrt{841}}{8} = \frac{11 \pm 29}{8}
\][/tex]
This gives us two possible values for [tex]\(y\)[/tex]:
- [tex]\(y = \frac{11 + 29}{8} = \frac{40}{8} = 5\)[/tex]
- [tex]\(y = \frac{11 - 29}{8} = \frac{-18}{8} = -\frac{9}{4}\)[/tex]
5. Back-Substitute for [tex]\(x\)[/tex]:
Remember [tex]\(y = x^2\)[/tex], so substitute back:
- For [tex]\(y = 5\)[/tex], we have [tex]\(x^2 = 5\)[/tex]. This gives:
- [tex]\(x = \pm \sqrt{5}\)[/tex]
- For [tex]\(y = -\frac{9}{4}\)[/tex], we have [tex]\(x^2 = -\frac{9}{4}\)[/tex]. Since this is negative, the solutions will be complex:
- [tex]\(x = \pm \sqrt{-\frac{9}{4}} = \pm \frac{3}{2}i\)[/tex]
6. Final Solutions:
The complete solution set for the original polynomial equation is:
[tex]\[
x = -\sqrt{5}, \sqrt{5}, -\frac{3}{2}i, \frac{3}{2}i
\][/tex]
These are the four roots of the equation, including two real and two imaginary solutions.
