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Magazine Chapter 11: Problem 38
Solve. \(x^{2 / 3}-2 x^{1 / 3}-8=0\)
Short Answer
Expert verified
The solutions for \( x \) are \( 64 \) and \( -8 \).
Step by step solution
01
Set up Substitution
First, let's substitute a new variable to simplify the equation. Let \( y = x^{1/3} \). This means \( y^2 = (x^{1/3})^2 = x^{2/3} \). Substituting in, the equation becomes \( y^2 - 2y - 8 = 0 \).
02
Solve Quadratic Equation
The equation \( y^2 - 2y - 8 = 0 \) is a quadratic equation. We can solve it using the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -2 \), and \( c = -8 \). Substitute these values in: \( y = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \).
03
Calculate Discriminant
Calculate the discriminant: \((b^2 - 4ac)\). Here, \((-2)^2 = 4\) and \(4 \cdot 1 \cdot (-8) = -32\). Therefore, the discriminant is \(4 + 32 = 36\).
04
Compute Roots of the Quadratic
Now calculate the roots: \( y = \frac{2 \pm \sqrt{36}}{2} \). The square root of 36 is 6, so \( y = \frac{2 \pm 6}{2} \). Hence, the solutions for \( y \) are \( y = \frac{8}{2} = 4 \) and \( y = \frac{-4}{2} = -2 \).
05
Reverse Substitution
Recall that \( y = x^{1/3} \). So \( x^{1/3} = 4 \) and \( x^{1/3} = -2 \).
06
Solve for \( x \)
Raise both sides to the power of 3 to solve for \( x \). For \( x^{1/3} = 4 \), we get \( x = 4^3 = 64 \). For \( x^{1/3} = -2 \), we get \( x = (-2)^3 = -8 \). Thus the solutions are \( x = 64 \) and \( x = -8 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
Quadratic equations are a type of polynomial equation of the form \( ax^2 + bx + c = 0 \). They play a crucial role in algebra and allow us to model many real-world scenarios. The general form of a quadratic equation involves:
- \( a \): the coefficient of the squared term \( x^2 \)
- \( b \): the coefficient of the linear term \( x \)
- \( c \): the constant term
Substitution Method
The substitution method is a technique used to simplify complex equations by substituting variables. This can make equations easier to handle, especially when working with higher degree polynomials or complex expressions. In the given problem:
- The substitution \( y = x^{1/3} \) was used, which converts the original equation \( x^{2/3}-2x^{1/3}-8=0 \) into a simpler quadratic form \( y^2 - 2y - 8 = 0 \).
Discriminant
The discriminant is a part of the quadratic formula, crucial for determining the nature and number of solutions of a quadratic equation. It is given by the expression \( b^2 - 4ac \). When investigating a quadratic equation:
- If the discriminant is positive, the equation has two distinct real roots.
- If the discriminant equals zero, there is exactly one real root, often referred to as a repeated or double root.
- If the discriminant is negative, the equation has two complex roots.
Cubic Roots
Cubic roots are the inverse operation of cubing a number. If you have a number \( x \) and you find that its cube is \( y \), then \( x \) is a cubic root of \( y \). Solving equations involving cubic roots often requires conversion into a simpler form, such as through a substitution method.In this exercise, after solving the simpler quadratic equation for \( y \), we obtained \( y = 4 \) and \( y = -2 \). Since \( y \) was defined as \( x^{1/3} \), reversing the substitution gives us:
- For \( x^{1/3} = 4 \), cube both sides to get \( x = 4^3 = 64 \).
- For \( x^{1/3} = -2 \), cube both sides to find \( x = (-2)^3 = -8 \).
