Problem 14 Solve. $$t^{4}-11 t^{2}+18=0$$... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 8: Problem 14

Solve. $$t^{4}-11 t^{2}+18=0$$

Short Answer

Expert verified

The solutions are $ t = \pm 3 $ and $ t = \pm \sqrt{2} $.

Step by step solution

Substitute to transform the equation

Let us set a substitution to simplify the equation. Substitute $ u = t^{2} $. Thus, the equation becomes $ u^{2} - 11u + 18 = 0 $.

Solve the quadratic equation

Now, solve the quadratic equation for $ u $. Factoring the quadratic equation, we get $(u - 9)(u - 2) = 0 $. Therefore, $ u = 9 $ or $ u = 2 $.

Substitute back to find t

Replace $ u $ back with $ t^{2} $: $ t^{2} = 9 $ and $ t^{2} = 2 $. Solving these, we get $ t = \pm 3 $ and $ t = \pm \sqrt{2} $.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Substitution

Quadratic substitution helps simplify complex polynomial equations. By using substitution, you turn a higher-degree polynomial into a quadratic equation, which is easier to solve. In our exercise, we start with a fourth-degree polynomial: $t^{4}-11 t^{2}+18=0$. This looks complicated, so we set a substitution: $u = t^{2}$. This converts the original equation into a simpler quadratic equation: $u^{2} - 11u + 18 = 0$. This new equation is much easier to handle.

Factoring Quadratic Equations

Once you have a quadratic equation, the next step is to solve it. One common method is factoring. Factoring means rewriting the quadratic equation as a product of two simpler expressions. For the equation $u^{2} - 11u + 18 = 0$, we look for two numbers that multiply to 18 and add up to -11. These numbers are -9 and -2. So, we rewrite the equation as $ (u - 9)(u - 2) = 0 $. This gives us two possible solutions: $u = 9 $ or $u = 2$.

Solving Quadratic Equations

After factoring, solving the quadratic equation is straightforward. Each factor gives us a potential solution for $u$. In our example, we found $u = 9$ and $u = 2$. But remember, $u$ was a substitution for $t^{2}$. So we need to replace $u$ back with $t^{2}$. This gives us: $t^{2} = 9$ and $t^{2} = 2$. For $t^{2} = 9$, we take the square root to find $t = \pm 3$. For $t^{2} = 2$, we get $t = \pm \sqrt{2}$. So, the complete solutions to our original equation are $t = 3, t = -3, t =\pm \sqrt{2}$.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve. $$t^{4}-11 t^{2}+18=0$$

Short Answer

Step by step solution

Substitute to transform the equation

Solve the quadratic equation

Substitute back to find t

Key Concepts

Quadratic Substitution

Factoring Quadratic Equations

Solving Quadratic Equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.