Problem 92 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ \frac{1}{4} x^{2}+3 x=x-4 $$

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 + 8x + 16 = 0\), with one real solution \(x = -4\).

Step 1: Rearrange the equation

Start with the given equation \( \frac{1}{4}x^{2} + 3x = x - 4 \). Move all terms to one side to get it in the form \( ax^{2} + bx + c = 0 \). Subtract \(x\) and add \(4\) to both sides to obtain: \( \frac{1}{4}x^{2} + (3x - x) + 4 = 0 \), which simplifies to \( \frac{1}{4}x^{2} + 2x + 4 = 0 \).

Step 2: Convert to standard quadratic form

To make \(a > 0\) and to eliminate the fraction in \(a\), multiply the entire equation by 4 to clear the fraction: \(4 \cdot \left(\frac{1}{4}x^{2} + 2x + 4\right) = 4 \cdot 0\). This results in the equation \(x^{2} + 8x + 16 = 0\). Here, \(a = 1\), \(b = 8\), and \(c = 16\).

Step 3: Calculate the discriminant

The discriminant is calculated using the formula \(b^{2} - 4ac\). Substitute the values into the formula: \(8^{2} - 4 \cdot 1 \cdot 16\). Simplifying gives \(64 - 64 = 0\). A discriminant of zero means there is exactly one real solution.

Step 4: Solve the equation using the quadratic formula

Since the discriminant is zero, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Substitute the values: \(x = \frac{-8 \pm \sqrt{0}}{2 \cdot 1}\). This simplifies to \(x = \frac{-8}{2}\), resulting in \(x = -4\). Hence, there is one real solution \(x = -4\).

Key Concepts

Discriminant

In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is a key mathematical expression that helps determine the nature of the solutions. The discriminant is calculated using the formula \( b^2 - 4ac \). It gives us insight into the type of solutions we can expect without directly solving the equation.

Here are some important insights regarding the discriminant:

• If the discriminant is greater than zero \( (b^2 - 4ac > 0) \), there are two distinct real solutions.
• If the discriminant is exactly zero \( (b^2 - 4ac = 0) \), there is exactly one real solution, sometimes referred to as a repeated or double root.
• If the discriminant is less than zero \( (b^2 - 4ac < 0) \), there are no real solutions, but instead, the solutions are complex conjugates.

In our solution, the discriminant is calculated as \( 64 - 64 = 0 \), indicating a single real solution.

Real Solutions

Real solutions of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These solutions are derived based on the value of the discriminant.

When solving quadratic equations, determining the number of real solutions is a crucial step:

• With a discriminant greater than zero, you would have two real and distinct solutions.
• A discriminant of zero results in exactly one real solution. This occurs because the quadratic curve just touches the x-axis at the vertex of the parabola.
• With a discriminant less than zero, the parabola does not intersect the x-axis, implying imaginary or complex solutions.

In the given exercise, the discriminant is zero. This tells us that there is one real solution, making the process straightforward without needing to consider complex solutions.

Quadratic Formula

The quadratic formula is an essential tool for finding the solutions to quadratic equations. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This formula provides a direct way to calculate the solutions to any quadratic equation, assuming the values of \(a\), \(b\), and \(c\) are known:

- \(-b\) indicates the opposite sign of the coefficient \(b\).- The plus-and-minus symbol (\( \pm \)) symbolizes that the quadratic equation will have two solutions - the sum and the difference of the square root of the discriminant.- \(2a\) represents twice the value of the coefficient \(a\), which scales the solution relative to \(a\).

In our exercise, because the discriminant is zero, we use the formula:\[x = \frac{-8 \pm \sqrt{0}}{2 \cdot 1}\]This simplifies to \(x = -4\), confirming there is only one real solution. The quadratic formula shows robust utility in solving any quadratic equation, making it a fundamental concept in algebra.