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Chapter 8: Problem 33

Solve. $$ t^{2 / 3}+t^{1 / 3}-6=0 $$

Short Answer

Expert verified
The solutions are t = 8 and t = -27.

Step by step solution

01

Identify Substitution

Let's simplify the equation by substituting a new variable. Let us set \[ u = t^{1/3} \] Then the equation becomes: \[ u^2 + u - 6 = 0 \]
02

Solve the Quadratic Equation

We now solve the quadratic equation \( u^2 + u - 6 = 0 \). This can be factored as: \[ (u - 2)(u + 3) = 0 \]. Therefore, we have: \[ u - 2 = 0 \] or \[ u + 3 = 0 \] which gives the solutions \[ u = 2 \] or \[ u = -3 \]
03

Substitute Back

Recall that \( u = t^{1/3} \). We substitute back to find the values of \( t \): \[ t^{1/3} = 2 \] which yields \[ t = 2^3 = 8 \] and \[ t^{1/3} = -3 \] which yields \[ t = (-3)^3 = -27 \]
04

Check Solutions

We check both solutions by plugging them back into the original equation:\( t = 8 \): \[ 8^{2/3} + 8^{1/3} - 6 = 4 + 2 - 6 = 0 \] This is correct.\( t = -27 \): \[ (-27)^{2/3} + (-27)^{1/3} - 6 = 9 - 3 - 6 = 0 \] This is also correct.

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Key Concepts

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

quadratic equation
A quadratic equation has the general form \( ax^2 + bx + c = 0 \). It is called 'quadratic' because 'quad' means square, referring to the variable鈥檚 highest exponent. In our exercise, the quadratic equation is the transformed equation \( u^2 + u - 6 = 0 \) after substitution.
This simplified form helps us solve the original problem easier.
Solving a quadratic equation typically involves three methods:
  • Factoring,
  • Using the quadratic formula,
  • Completing the square.
In this exercise, we use factoring to solve it. We factored \( u^2 + u - 6 \) into \( (u - 2)(u + 3) \), which lets us find the solutions for 'u'. Once we have 'u', we substitute back to get the original variable.
substitution method
The substitution method involves replacing a complex expression with a simpler variable, which makes the equation easier to handle. In our exercise, we started with the equation \( t^{2/3} + t^{1/3} - 6 = 0 \) and set \( u = t^{1/3} \). This turned our equation into \( u^2 + u - 6 = 0 \).
By using substitution:
  • We made the problem simpler and more recognizable.
  • The quadratic form was easier to solve.

After solving for 'u', we did a 'reverse substitution' to find the values for 't' by solving \( t^{1/3} = u \).
factoring
Factoring is one of the methods used to solve quadratic equations, by expressing the quadratic as a product of linear factors. For the equation \( u^2 + u - 6 = 0 \), our goal was to express it in the form of \( (u - a)(u + b) = 0 \). We found that the factors were \( u - 2 \) and \( u + 3 \), simplifying the equation to \( (u - 2)(u + 3) = 0 \).
This allowed us to easily identify the solutions: \( u = 2 \) and \( u = -3 \).
Factoring works by:
  • Finding two numbers that multiply to give the constant term (in our case '-6')
  • And adding to give the coefficient of the 'u' term (which is '1').
For our problem, factors of -6 that add up to 1 are 2 and -3, hence we factored it as \( (u - 2)(u + 3) \).

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Most popular questions from this chapter

For over 2000 years, artists, sculptors, and architects have regarded the proportions of a "golden" rectangle as visually appealing. A rectangle of width \(w\) and length \(l\) is considered "golden" if $$ \frac{w}{l}=\frac{l}{w+l} $$ Solve for \(l\).

Find a number for which the reciprocal of 1 less than the number is the same as 1 more than the number.

Use a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$f(x)=x^{3}-2 x^{2}-5 x+6$$

Rational Inequalities Solve. For \(F(x)=\frac{1}{x-3},\) find all \(x\) -values for which \(F(x) \leq 2\).

Explain how any quadratic inequality can be solved by examining a parabola.
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