Problem 10 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. Write your solutions in simplest form. $$4 x^{2}-8 x+3=0$$

Step-by-Step Solution

Answer

The roots of the equation \(4x^{2} - 8x + 3 = 0\) are \(x_{1} = 1.5\) and \(x_{2} = 0.5\).

Step 1: Identify the coefficients

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

Step 2: Substitute the coefficients into the quadratic formula

The quadratic formula is given as \( x = [-b \pm sqrt(b^{2} - 4ac)] / 2a \). Substitute the coefficients a=4, b=-8, and c=3 into the quadratic formula we get: \( x = [8 \pm sqrt((-8)^{2} - 4 * 4 * 3)] / (2 * 4) \).

Step 3: Simplify the equation

First, compute the discriminant \( b^{2} - 4ac \) = (-8)^{2} - 4 * 4 * 3 = 64 - 48 = 16. Replace the discriminant into the equation: \( x = [8 \pm sqrt(16)] / 8 = (8 \pm 4) / 8 \). This gives two solutions: \( x = 12/8 \) or \( x = 4/8 \).

Step 4: Simplify each solution

The given solutions can be simplified to their lowest terms. There purfore \( x_{1} = 1.5 (12/8) \) and \( x_{2} = 0.5 (4/8) \).

Key Concepts

Quadratic Equation

A quadratic equation is a mathematical expression that can be represented in the form \( ax^2 + bx + c = 0 \). It involves terms where the variable (usually \( x \)) is raised to the second power. Here are some key points about quadratic equations:

• General Form: The general form of a quadratic equation is where \( a \), \( b \), and \( c \) are coefficients that can be any real number, with the condition \( a eq 0 \).
• Degree: This type of equation is called "quadratic" because it involves the square of the unknown variable (\( x^2 \)), hence it is a second-degree polynomial.
• Purpose: Solving a quadratic equation involves finding the values of \( x \) that satisfy the equation, known as "roots" or "solutions."

In the example given from the exercise, the equation is \( 4x^2 - 8x + 3 = 0 \). Our task is to find the values of \( x \) that make this equation true using the quadratic formula. This approach offers an efficient way to solve any quadratic equation.

Discriminant

The discriminant is a specific component of the quadratic formula. It gives us valuable information about the roots of a quadratic equation. The discriminant is defined as \( b^2 - 4ac \). Let's explore its role and importance:

• Purpose: The discriminant helps determine the nature and number of roots for a quadratic equation without actually solving it.
• Positive Discriminant: If the discriminant is positive, the quadratic equation has two distinct real roots.
• Zero Discriminant: A discriminant of zero signifies that there is exactly one real root, indicating a perfect square trinomial.
• Negative Discriminant: If the discriminant is negative, no real roots exist; instead, the equation has two complex roots.

In the exercise's equation \( 4x^2 - 8x + 3 = 0 \), the discriminant was calculated as 16, which is positive. This tells us there are two distinct real roots, which the quadratic formula will provide.

Coefficients

Coefficients are the numerical factors in terms of variables within an equation. For a quadratic equation \( ax^2 + bx + c = 0 \), the coefficients help dictate the shape and position of its graph, among other properties:

• Leading Coefficient \( a \): It multiplies the \( x^2 \) term and influences the parabola's direction. If \( a \) is positive, the parabola opens upwards. If negative, it opens downwards.
• Linear Coefficient \( b \): This influences the parabola's position along the x-axis. Changes in \( b \) translate the vertex horizontally.
• Constant Term \( c \): This term represents the y-intercept (where the parabola crosses the y-axis).

For the equation \( 4x^2 - 8x + 3 = 0 \) from the exercise, the coefficients are \( a = 4 \), \( b = -8 \), and \( c = 3 \). These values are crucial because substituting them into the quadratic formula allows us to find the precise solutions for \( x \). Understanding each coefficient’s role gives us insight into how changes in numbers affect the equation's properties and solutions.