Problem 25 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$\frac{1}{3} t^{2}-t+\frac{1}{6}=0$$

Step-by-Step Solution

Answer

The solutions are \( t = \frac{3 + \sqrt{7}}{2}\) and \( t = \frac{3 - \sqrt{7}}{2}\).

Step 1: Identify the coefficients

The given equation is \(\frac{1}{3} t^{2}-t+\frac{1}{6}=0\). Identify the coefficients from the standard quadratic form \(ax^2 + bx + c = 0\). Here, \(a = \frac{1}{3}\), \(b = -1\), and \(c = \frac{1}{6}\).

Step 2: Write down the quadratic formula

The quadratic formula is \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Substitute the coefficients into the quadratic formula

Substitute \(a = \frac{1}{3}\), \(b = -1\), and \(c = \frac{1}{6}\) into the quadratic formula: \(t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 (\frac{1}{3})(\frac{1}{6})}}{2 (\frac{1}{3})}\).

Step 4: Simplify the expression inside the square root

Calculate the discriminant: \$ \(\sqrt{(-1)^2 - 4 \cdot \frac{1}{3} \cdot \frac{1}{6}} = \[ \sqrt{1 - \frac{4}{18}} = \sqrt{1 - \frac{2}{9}} = \sqrt{\frac{9}{9} - \frac{2}{9}} = \sqrt{\frac{7}{9}}\] = \frac{\sqrt{7}}{3}\).

Step 5: Simplify the entire expression

Combine all the parts: \( t = \frac{1 \pm \sqrt{\frac{7}{9}}}{\frac{2}{3}} = \frac{1 \pm \frac{\sqrt{7}}{3}}{\frac{2}{3}} = \frac{1 \pm \frac{\sqrt{7}}{3}}{\frac{2}{3}} = \frac{1 \pm \frac{\sqrt{7}}{3}} \cdot \frac{3}{2} = \frac{3}{2} \pm \frac{\sqrt{7}}{2}\).

Step 6: Write the final solutions

The solutions are \( t = \frac{3 + \sqrt{7}}{2}\) and \( t = \frac{3 - \sqrt{7}}{2}\).

Key Concepts

Solving Quadratic Equations

When solving quadratic equations, the quadratic formula is your go-to tool. It helps you find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula to remember is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Using this formula involves several steps, including substituting the right values and simplifying the results.

Coefficients Identification

Identifying the coefficients is your first step in using the quadratic formula. These coefficients are the constants that multiply each term in the equation. For a quadratic equation \(ax^2 + bx + c = 0\), you'll need to pinpoint \(a\), \(b\), and \(c\). In the given equation \(\frac{1}{3} t^{2}-t+\frac{1}{6}=0\), the coefficients are:

• \(a = \frac{1}{3}\)
• \(b = -1\)
• \(c = \frac{1}{6}\)

These values are essential for the next steps of solving the equation, so it’s important to identify them correctly.

Discriminant Calculation

The discriminant inside the quadratic formula, \(b^2 - 4ac\), reveals the nature of the roots. It helps determine whether you will have real or complex solutions. Here’s how to calculate it: For the equation \(\frac{1}{3} t^{2}-t+\frac{1}{6}=0\), the discriminant is calculated as follows:
\[\sqrt{(-1)^2 - 4 \cdot \frac{1}{3} \cdot \frac{1}{6}} = \sqrt{1 - \frac{4}{18}} = \sqrt{1 - \frac{2}{9}} = \sqrt{\frac{9}{9} - \frac{2}{9}} = \sqrt{\frac{7}{9}} = \frac{\sqrt{7}}{3}\] The simplified discriminant \( \frac{\sqrt{7}}{3} \) informs us that the roots are real and distinct.

Simplifying Square Roots

Simplifying square roots is crucial in solving quadratic equations with the quadratic formula. It makes your computation easier and the final answers clearer. Here, the discriminant simplified to \( \frac{\sqrt{7}}{3} \). This value needs to be further simplified within the quadratic formula:

• First, write the discriminant as \( \frac{\sqrt{7}}{3} \)
• Combine it with the rest of the formula: \( t = \frac{1 \pm \frac{\sqrt{7}}{3}}{\frac{2}{3}} \)

Having a clear, simplified discriminant helps in accurately finding the roots.

Fraction Operations

Dealing with fractions is often part of the process when using the quadratic formula. Accurate fraction operations are essential to reach the correct solutions. Here’s a quick guide using our equation:

• Step 1: Combine the terms: \(t = \frac{1 \pm \frac{\sqrt{7}}{3}}{\frac{2}{3}}\)
• Step 2: Flip and multiply the divisor: \(t = (1 \pm \frac{\sqrt{7}}{3}) \times \frac{3}{2}\)
• Step 3: Distribute the fraction: \(t = \frac{3}{2} \pm \frac{\sqrt{7}}{2}\)

We end up with two possible solutions: \(t = \frac{3 + \sqrt{7}}{2}\) and \(t = \frac{3 - \sqrt{7}}{2}\). Each step involves careful handling of the fractions to ensure accuracy.