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Chapter 9: Problem 156

Solve. $$x^{4}-9 x^{2}+18=0$$

Short Answer

Expert verified
The solutions are \( x = \pm \sqrt{3} \) and \( x = \pm \sqrt{6} \).

Step by step solution

01

Identify the Quadratic Form

Notice that the given equation can be rewritten in the form of a quadratic by substituting another variable. Let \( y = x^2 \), then the equation becomes \( y^2 - 9y + 18 = 0 \).
02

Solve the New Quadratic Equation

Now solve the transformed quadratic equation for \( y \). This can be factored as \( (y - 3)(y - 6) = 0 \). Therefore, the solutions for \( y \) are \( y = 3 \) and \( y = 6 \).
03

Back-Substitute \( y \) with \( x^2 \)

Replace \( y \) with \( x^2 \) in the solutions. So, you get two equations: \( x^2 = 3 \) and \( x^2 = 6 \).
04

Solve for \( x \)

Solve each equation for \( x \): \( x^2 = 3 \) gives \( x = \pm \sqrt{3} \), and \( x^2 = 6 \) gives \( x = \pm \sqrt{6} \).
05

Combine All Solutions

The complete set of solutions for the original equation \( x^4 - 9x^2 + 18 = 0 \) is \( x = \pm \sqrt{3} \) and \( x = \pm \sqrt{6} \).

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Key Concepts

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

quadratic substitution
Quadratic substitution is a powerful technique used to simplify complex polynomial equations by transforming them into a more familiar quadratic form.
This is especially useful when dealing with quartic equations, which can often seem daunting at first.
In the example problem, we start with the equation \( x^4 - 9x^2 + 18 = 0 \).
factoring polynomials
Factoring polynomials is a fundamental skill in algebra.
It involves breaking down a polynomial into simpler components that, when multiplied together, give you the original polynomial.
In this case, we need to factor the quadratic equation we obtained from our substitution: \( y^2 - 9y + 18 = 0 \).
solving quadratic equations
Once a quadratic equation is factored, solving it is simple.
Each factor gives us a possible solution for the equation.
From \( (y - 3)(y - 6) = 0 \), we find two solutions: \( y = 3 \) and \( y = 6 \).
square roots
Square roots play a crucial role in solving polynomial equations, especially when dealing with equations involving squared terms.
In our case, after substituting and solving the quadratic, we arrived at square root expressions: \( x^2 = 3 \) and \( x^2 = 6 \).

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Most popular questions from this chapter

Solve each inequality algebraically and write any solution in interval notation. $$2 x^{2}+5 x-12>0$$

Graph each function. $$f(x)=-4 x^{2}$$

Solve each inequality algebraically and write any solution in interval notation. $$x^{2}-7 x+10<0$$

Graph each function using a vertical shift. $$f(x)=x^{2}+3$$

(a) graph the quadratic functions on the same rectangular coordinate system and (b) describe what effect adding a constant, \(h\), inside the parentheses has \(f(x)=x^{2}, g(x)=(x-3)^{2},\) and \(h(x)=(x+3)^{2}\)
See all solutions

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