Find detailed and accurate answers to your questions on IDNLearn.com. Ask any question and get a detailed, reliable answer from our community of experts.

What are the solutions of the equation [tex]x^4 + 95x^2 - 500 = 0[/tex]? Use factoring to solve.

A. [tex]x = \pm \sqrt{5}[/tex] and [tex]x = \pm 10[/tex]

B. [tex]x = \pm i \sqrt{5}[/tex] and [tex]x = \pm 10i[/tex]

C. [tex]x = \pm \sqrt{5}[/tex] and [tex]x = \pm 10i[/tex]

D. [tex]x = \pm i \sqrt{5}[/tex] and [tex]x = \pm 10[/tex]


Sagot :

To solve the equation [tex]\( x^4 + 95x^2 - 500 = 0 \)[/tex] using factoring, we can start by making a substitution to simplify our approach. Let's set [tex]\( y = x^2 \)[/tex]. This transforms the original equation into a quadratic equation:
[tex]\[ y^2 + 95y - 500 = 0 \][/tex]

Now, we need to solve the quadratic equation [tex]\( y^2 + 95y - 500 = 0 \)[/tex]. To do this, we can either factor it (if possible) or use the quadratic formula. We'll use the quadratic formula in this case:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

For our equation [tex]\( y^2 + 95y - 500 = 0 \)[/tex], the coefficients are:
[tex]\[ a = 1, \quad b = 95, \quad c = -500 \][/tex]

Plugging these values into the quadratic formula gives:
[tex]\[ y = \frac{-95 \pm \sqrt{95^2 - 4 \cdot 1 \cdot (-500)}}{2 \cdot 1} \][/tex]
[tex]\[ y = \frac{-95 \pm \sqrt{9025 + 2000}}{2} \][/tex]
[tex]\[ y = \frac{-95 \pm \sqrt{11025}}{2} \][/tex]
[tex]\[ y = \frac{-95 \pm 105}{2} \][/tex]

This results in two solutions:
[tex]\[ y = \frac{-95 + 105}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ y = \frac{-95 - 105}{2} = \frac{-200}{2} = -100 \][/tex]

Now we revert back to our original variable [tex]\( x \)[/tex]. Recall that [tex]\( y = x^2 \)[/tex], so we now have:
[tex]\[ x^2 = 5 \quad \text{or} \quad x^2 = -100 \][/tex]

For [tex]\( x^2 = 5 \)[/tex], solving for [tex]\( x \)[/tex] gives:
[tex]\[ x = \pm \sqrt{5} \][/tex]

For [tex]\( x^2 = -100 \)[/tex], solving for [tex]\( x \)[/tex] gives:
[tex]\[ x = \pm \sqrt{-100} = \pm 10i \][/tex]
(where [tex]\( i \)[/tex] is the imaginary unit, [tex]\( i^2 = -1 \)[/tex]).

Therefore, the solutions to the equation [tex]\( x^4 + 95x^2 - 500 = 0 \)[/tex] are:
[tex]\[ x = \pm \sqrt{5} \text{ and } x = \pm 10i \][/tex]

Among the provided choices, the correct one is:
[tex]\[ x = \pm \sqrt{5} \text{ and } x = \pm 10i \][/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover insightful answers at IDNLearn.com. We appreciate your visit and look forward to assisting you again.