Problem 55 - Intermediate Algebra

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

Step 1: Identify the Equation Type

The given equation is a quadratic equation in the standard form, \(4x^2 - 8x + 3 = 0\). The coefficients are \(a = 4\), \(b = -8\), and \(c = 3\).

Step 2: Check for Factoring Possibility

We first check if the quadratic is easily factorable. We look for two numbers that multiply to \(ac = 4 \times 3 = 12\) and add up to \(b = -8\). However, such numbers do not exist. Thus, factoring is not possible here.

Step 3: Use the Quadratic Formula

As factoring is not straightforward, we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac = (-8)^2 - 4 \times 4 \times 3 = 64 - 48 = 16\).

Step 5: Solve Using the Quadratic Formula

Substitute the values into the quadratic formula: \[x = \frac{-(-8) \pm \sqrt{16}}{2 \times 4}\]Simplify: \[x = \frac{8 \pm 4}{8}\].

Step 6: Evaluate the Expressions

Now solve for the two possible values of \(x\).1. \(x = \frac{8 + 4}{8} = \frac{12}{8} = \frac{3}{2}\)2. \(x = \frac{8 - 4}{8} = \frac{4}{8} = \frac{1}{2}\).

Step 7: Conclusion

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

Key Concepts

Quadratic Formula

One of the most effective methods for solving quadratic equations is the quadratic formula. The quadratic formula is specifically designed to find solutions (also known as "roots" or "zeros") for any quadratic equation, which takes the standard form: \[ ax^2 + bx + c = 0 \] The formula itself is: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \] Where:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

To use the quadratic formula, you simply substitute the values of \(a\), \(b\), and \(c\) into the formula and solve for \(x\). This formula is particularly useful when a quadratic equation is not easily factored, or when you want an exact, reliable method. This allows us to find the solutions even for equations that do not neatly resolve through other methods.

Discriminant

The discriminant is a part of the quadratic formula that provides essential information about the roots of a quadratic equation even before solving it. It is given by the expression under the square root in the quadratic formula: \( b^2 - 4ac \).The discriminant determines the nature and number of roots as follows:

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (also known as a repeated or double root).
• If \(b^2 - 4ac < 0\), the equation has two complex roots (imaginary roots) with no real solutions.

In our exercise, the discriminant was calculated as 16, which is greater than zero, indicating the presence of two distinct real roots. Understanding the discriminant helps us predict the solutions we might find when using the quadratic formula or other solving techniques.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. Quadratic equations are second-degree polynomials, and solving them can be done through several methods, but the quadratic formula is one of the most universal.To begin solving a quadratic equation, you first need to ensure it's set to zero, like this: \[ ax^2 + bx + c = 0 \]Let's briefly touch upon possible methods:

• Factoring: Attempt to express the equation as a product of binomials. This works best when the equation is easily factorable.
• Quadratic Formula: Use this formula to find solutions when factoring is difficult or impossible.
• Completing the Square: This method transforms the quadratic into a perfect square trinomial.

In our specific problem, \(4x^2 - 8x + 3 = 0\), using the quadratic formula was ideal since factoring wasn't an easy route. After calculating the discriminant and confirming the type of roots, substituting into the quadratic formula gave us the solutions, \(x = \frac{3}{2}\) and \(x = \frac{1}{2}\). Each step in solving these equations helps build a deeper understanding and ease for tackling more complex mathematical challenges.