Problem 55 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate. $$4 x^{2}-8 x+3=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\).

Step 1: Identify the Quadratic Equation

The given quadratic equation is \(4x^2 - 8x + 3 = 0\). It is a quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a = 4\), \(b = -8\), and \(c = 3\).

Step 2: Check the Discriminant

To determine the best method to solve the equation, calculate the discriminant \(D\) using the formula \(D = b^2 - 4ac\). Substitute the given values: \(D = (-8)^2 - 4 imes 4 imes 3 = 64 - 48 = 16\).

Step 3: Determine the Solution Method

Since the discriminant \(D = 16\) is a perfect square, the equation has two distinct real roots and factoring is appropriate. Additionally, the quadratic formula can handle perfect squares easily. We can proceed with either method.

Step 4: Choose Factoring Method

Let's proceed with factoring. First, rewrite the quadratic as \(4x^2 - 8x + 3 = 0\). We need to find two numbers that multiply to \(4 \times 3 = 12\) and add to \(-8\). These numbers are \(-6\) and \(-2\).

Step 5: Rewrite and Factor

Rewrite the equation: \(4x^2 - 6x - 2x + 3 = 0\). Group the terms: \((4x^2 - 6x) + (-2x + 3) = 0\). Factor by grouping: \(2x(2x - 3) - 1(2x - 3) = 0\). This gives \((2x - 1)(2x - 3) = 0\).

Step 6: Solve for Roots

Set each factor equal to zero: 1. Solve \(2x - 1 = 0\): - Add 1 to both sides: \(2x = 1\). - Divide both sides by 2: \(x = \frac{1}{2}\).2. Solve \(2x - 3 = 0\): - Add 3 to both sides: \(2x = 3\). - Divide both sides by 2: \(x = \frac{3}{2}\).

Step 7: Conclude the Solutions

The solutions to the quadratic equation \(4x^2 - 8x + 3 = 0\) are \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\).

Key Concepts

Discriminant

The discriminant is an important tool in solving quadratic equations, and it helps us determine the number and type of roots we might expect. When you have a quadratic equation in the standard form of \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated by the formula:

• \( D = b^2 - 4ac \)

In our example, the quadratic equation is \( 4x^2 - 8x + 3 = 0 \). By substituting \( a = 4 \), \( b = -8 \), and \( c = 3 \), we found that the discriminant is \( D = 16 \).
Here are some key points:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root.
• If \( D < 0 \), no real roots exist; instead, there are two complex roots.

Since \( D = 16 \) is a positive value and a perfect square in our problem, we expect two distinct real roots. This suggests that both factoring and the quadratic formula could be used effectively to find the solution. Calculating the discriminant first is a great way to decide which method to pursue.

Factoring

Factoring involves rewriting the quadratic equation to express it as a product of simpler binomials. This method is often used when the discriminant is a perfect square because it tends to simplify calculations. In our exercise, we have the quadratic equation \( 4x^2 - 8x + 3 = 0 \), and we look for two numbers that multiply to \( ac \) (where \( a = 4 \) and \( c = 3 \), so \( ac = 12 \)) and add to \( b \) (which is \(-8\)).
The numbers \(-6\) and \(-2\) satisfy these conditions, so we can rewrite the middle term and factor by grouping:

• Rewrite the equation as \( 4x^2 - 6x - 2x + 3 = 0 \)
• Group the terms: \((4x^2 - 6x) + (-2x + 3) = 0\)
• Factor both groups: \(2x(2x - 3) - 1(2x - 3) = 0\)
• The factored equation is \((2x - 1)(2x - 3) = 0\)

Once factored, you can solve for the roots by setting each factor to zero. This gives us the solutions \( x = \frac{1}{2} \) and \( x = \frac{3}{2} \).
Factoring is a quick and effective method when the discriminant is a perfect square, providing straightforward solutions without the need for more complex calculations.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation. It is especially helpful when the quadratic does not easily factor. The formula itself is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula directly utilizes the discriminant \( b^2 - 4ac \) under the square root. For the given equation \( 4x^2 - 8x + 3 = 0 \), we've already calculated the discriminant \( D = 16 \).
Substituting \( a = 4 \), \( b = -8 \), and \( c = 3 \) into the quadratic formula:

• \( x = \frac{-(-8) \pm \sqrt{16}}{2 \times 4} \)
• \( x = \frac{8 \pm 4}{8} \)

This computation yields the roots \( x = \frac{1}{2} \) and \( x = \frac{3}{2} \), aligning with the solutions we found through factoring.
The quadratic formula is powerful because it will always bring you to the solution, regardless of whether factors are readily apparent, and is perfect when other methods seem too complex or tricky.