Problem 65 - Beginning and Intermediate Algebra

Question

Solve each equation. Check the solutions. \(4 x^{4 / 3}-13 x^{2 / 3}+9=0\)

Step-by-Step Solution

Answer

The solutions are \( x \approx 3.375 \) and \( x = 1 \).

Step 1: Substitution

Let us substitute a new variable to simplify the equation. Set \( y = x^{2/3} \). Thus, the equation becomes \( 4y^2 - 13y + 9 = 0 \).

Step 2: Solve the Quadratic Equation

Solve the quadratic equation \( 4y^2 - 13y + 9 = 0 \) using the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4 \), \( b = -13 \), and \( c = 9 \).

Step 3: Apply the Quadratic Formula

Plug in the values: \[ y = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4} \] This simplifies to \[ y = \frac{13 \pm \sqrt{169 - 144}}{8} \] \[ y = \frac{13 \pm \sqrt{25}}{8} \] \[ y = \frac{13 \pm 5}{8} \].

Step 4: Solve for y

Solve the expressions: \( y = \frac{13 + 5}{8} = \frac{18}{8} = \frac{9}{4} \) and \( y = \frac{13 - 5}{8} = \frac{8}{8} = 1 \). So, we have two solutions for \( y \): \( y = \frac{9}{4} \) and \( y = 1 \).

Step 5: Re-substitute

Re-substitute \( y = x^{2/3} \): For \( y = \frac{9}{4} \), we have \( x^{2/3} = \frac{9}{4} \). For \( y = 1 \), we have \( x^{2/3} = 1 \).

Step 6: Solve for x

Solve for \( x \) in each case: \( x = \left( \frac{9}{4} \right)^{3/2} \) and \( x = 1^{3/2} \).Calculate the expressions: \[ x = \left( \frac{9}{4} \right)^{3/2} = \left( \frac{9}{4} \right)^{1.5} = \left( 2.25 \right)^{1.5} \approx 3.375 \] and \( x = 1 \). So, \( x \approx 3.375 \) and \( x = 1 \).

Step 7: Check the Solutions

Substitute \( x \approx 3.375 \) and \( x = 1 \) back into the original equation to check:For \(x \approx 3.375:\)\[ 4(3.375)^{4/3} - 13(3.375)^{2/3} + 9 \approx 0 \].For \( x = 1:\)\[ 4(1)^{4/3} - 13(1)^{2/3} + 9 = 0 \].Both solutions satisfy the original equation.

Key Concepts

Substitution Method

The substitution method is a powerful technique to simplify polynomial equations. It involves replacing a complex part of the equation with a new variable, making the equation easier to handle.

In our exercise, we start with the equation: \( 4x^{4/3} - 13x^{2/3} + 9 = 0 \).

To simplify it, we substitute \( y = x^{2/3} \), turning the original equation into a quadratic form: \( 4y^2 - 13y + 9 = 0 \).
This substitution reduces a higher-degree polynomial equation into a standard quadratic equation, which is much easier to solve.

• Substitution helps in identifying patterns and reducing complexity.
• Always revert your substitution to find the values of the original variable.

By simplifying equations, the substitution method allows for more straightforward solutions and clearer understanding.

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). They appear frequently in algebra and are essential for solving higher-degree polynomials by methods like substitution.

In our exercise, after substitution, the equation becomes \( 4y^2 - 13y + 9 = 0 \).
To solve this, we use the quadratic formula: \( y = \frac{-b \pm \sqrt {b^2 - 4ac}}{2a} \).
Plugging in our values of \( a = 4 \), \( b = -13 \), and \( c = 9 \):
The equation simplifies to: \( y = \frac{13 \pm \sqrt {169 - 144}}{8} \).
This further simplifies to: \( y = \frac{13 \pm 5}{8} \).
Solving for \( y \), we get the two values: \( y = \frac{9}{4} \) and \( y = 1 \).
These solutions to the quadratic equation are critical because they transform back into the original variable to solve the primary equation.

Checking Solutions

Checking your solutions is an important step in problem-solving to ensure they satisfy the original equation.

After solving for \( y \) and reverting our substitution, we get \( x \approx 3.375 \) and \( x = 1 \).

We must substitute these values back into the original equation to verify their correctness.

For \( x \approx 3.375 \):\( 4(3.375)^{4/3} - 13(3.375)^{2/3} + 9 \approx 0 \).
For \( x = 1 \):\( 4(1)^{4/3} - 13(1)^{2/3} + 9 = 0 \).

• Both solutions satisfy the original equation, confirming their correctness.
• Always remember to check solutions, as it validates your entire process.