Answer :
To find the sum of the solutions to the equation [tex]x^4 - 45x^2 + 324 = 0[/tex], we can use a substitution method. This equation is in the form of a quadratic in terms of [tex]y = x^2[/tex]. Let's make the substitution:
[tex]y^2 - 45y + 324 = 0[/tex]
Now, this is a quadratic equation in [tex]y[/tex], and 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 = -45[/tex], and [tex]c = 324[/tex]. Plugging these values into the formula, we get:
[tex]y = \frac{-(-45) \pm \sqrt{(-45)^2 - 4 \times 1 \times 324}}{2 \times 1}[/tex]
[tex]y = \frac{45 \pm \sqrt{2025 - 1296}}{2}[/tex]
[tex]y = \frac{45 \pm \sqrt{729}}{2}[/tex]
[tex]y = \frac{45 \pm 27}{2}[/tex]
This gives us two possible values for [tex]y[/tex]:
- [tex]y = \frac{45 + 27}{2} = 36[/tex]
- [tex]y = \frac{45 - 27}{2} = 9[/tex]
Now, we substitute back [tex]x^2 = y[/tex]:
For [tex]x^2 = 36[/tex], the solutions are [tex]x = \pm 6[/tex].
For [tex]x^2 = 9[/tex], the solutions are [tex]x = \pm 3[/tex].
Thus, the solutions for [tex]x[/tex] are [tex]6, -6, 3, -3[/tex].
The sum of the solutions can be calculated:
[tex]6 + (-6) + 3 + (-3) = 0[/tex].
Therefore, the sum of the solutions is [tex]0[/tex].
The correct multiple choice answer is A. 0.
