Problem 26 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$\frac{3}{4} x^{2}-2 x+\frac{1}{2}=0$$

Step-by-Step Solution

Answer

2.387 and 0.279

Step 1: Identify coefficients a, b, and c

In the given quadratic equation \(\frac{3}{4} x^2 - 2x + \frac{1}{2} = 0\), the coefficients are: \(a = \frac{3}{4}\), \(b = -2\), \(c = \frac{1}{2}\).

Step 2: Write down the quadratic formula

The quadratic formula to solve for \(x\) is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 3: Substitute the coefficients into the quadratic formula

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

Step 4: Simplify within the square root

Calculate within the square root: \((-2)^2 = 4\) and \(-4 \cdot \frac{3}{4} \cdot \frac{1}{2} = -1.5\). \[x = \frac{2 \pm \sqrt{4 - 1.5}}{1.5}\].

Step 5: Compute the discriminant

Simplify the expression under the square root: \[4 - 1.5 = 2.5\]. So we have: \[x = \frac{2 \pm \sqrt{2.5}}{1.5}\].

Step 6: Solve for the values of x

Solve for \(x\) by computing each part: \[x = \frac{2 + \sqrt{2.5}}{1.5}\] and \[x = \frac{2 - \sqrt{2.5}}{1.5}\].

Step 7: Simplify the solutions

Calculate the numerical values for the solutions: \[x = \frac{2 + 1.581}{1.5} \approx 2.387\] and \[x = \frac{2 - 1.581}{1.5} \approx 0.279\].

Key Concepts

solving quadratic equations

When dealing with quadratic equations, the quadratic formula is a powerful tool. A quadratic equation is of the form: \[ax^2 + bx + c = 0\] The quadratic formula solves for the variable \(x\) and is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] To use this formula, identify the coefficients \(a\), \(b\), and \(c\) from the equation. These coefficients are: \[a, b, c\]. Input these values into the formula to find the solutions for \(x\). Remember, the \(\pm\) symbol implies that there are usually two solutions: one by adding the \( \sqrt{b^2 - 4ac }\) and one by subtracting it.

discriminant

The discriminant is a key part of the quadratic formula contained within the square root: \[b^2 - 4ac\]. It determines the nature of the roots of the quadratic equation:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, also called a repeated or double root.
• If \(b^2 - 4ac < 0\), there are no real roots (the roots are complex or imaginary).

Calculating the discriminant is an important step as it informs you about the number and type of solutions you will get when solving the quadratic equation.

coefficients

Coefficients in a quadratic equation, denoted as \(a\), \(b\), and \(c\), are the constants that multiply the variable terms. In the generic form of a quadratic equation: \[ax^2 + bx + c = 0\]

• \(a\) is the coefficient of \(x^2\). It affects the width of the parabola when the quadratic is graphed.
• \(b\) is the coefficient of \(x\). It influences the location of the vertex along the x-axis.
• \(c\) is the constant term. It moves the parabola up or down on the graph.

Identifying these coefficients correctly is crucial when substituting them into the quadratic formula. In our example, \(a = \frac{3}{4}\), \(b = -2\), and \(c = \frac{1}{2}\).