Problem 8 - A Graphical Approach to Precalculus with Limits

Question

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Solve \(\frac{1}{4} x^{2}+x=1\).

Step-by-Step Solution

Answer

The solutions are \( x = -2 + 2\sqrt{2} \) and \( x = -2 - 2\sqrt{2} \).

Step 1: Set Equation to Zero

We start by rearranging the given equation to express it in the standard quadratic form, which is \( ax^2 + bx + c = 0 \). The original equation is \( \frac{1}{4}x^2 + x = 1 \). Subtract 1 from both sides to get: \( \frac{1}{4}x^2 + x - 1 = 0 \).

Step 2: Multiply to Eliminate Fractions

To eliminate the fraction, multiply every term by 4, the denominator of the fraction, resulting in: \( x^2 + 4x - 4 = 0 \).

Step 3: Use the Quadratic Formula

The equation \( x^2 + 4x - 4 = 0 \) is a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 4 \), and \( c = -4 \). Apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Step 4: Compute the Discriminant

Calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = 4^2 - 4\times1\times(-4) = 16 + 16 = 32 \].

Step 5: Solve for x using the Quadratic Formula

Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{32}}{2} \]. Simplify the square root and expression: \[ x = \frac{-4 \pm 4\sqrt{2}}{2} \]. Further simplify to get the solutions: \[ x = -2 \pm 2\sqrt{2} \].

Step 6: Write the Solutions

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

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions to any quadratic equation. A quadratic equation is typically written in the standard form: \( ax^2 + bx + c = 0 \). The quadratic formula itself is:

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

Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation. By plugging values into the formula, you can find the solutions for \( x \). This formula is particularly useful when factoring is difficult or impossible.
It's important to understand how each part of the formula works:

• \( -b \): This part reflects the symmetry of the roots around the vertical axis in the parabola.
• \( \pm \sqrt{...} \): The \( \pm \) sign indicates that there are generally two solutions, as squares have both positive and negative roots.
• \( 4ac \): Part of the discriminant, which we'll discuss in the next section.

Discriminant

The discriminant is a component of the quadratic formula, appearing under the square root sign: \( b^2 - 4ac \). Its value can tell us several important things about the nature of the solutions without even solving the equation.

• If the discriminant is positive, the quadratic equation has two distinct real roots. This indicates the parabola crosses the x-axis at two points.
• If the discriminant is zero, there is exactly one real root. This scenario happens when the vertex of the parabola just touches the x-axis.
• If the discriminant is negative, the equation has no real roots, which means the parabola does not cross the x-axis at all and solutions are complex numbers.

When solving \( x^2 + 4x - 4 = 0 \), you compute the discriminant as:\( 4^2 - 4 \times 1 \times (-4) = 32 \). Since 32 is positive, we have two real solutions. Understanding the discriminant gives insight into the problem and helps anticipate the type of solutions we will find.
Being aware of these possibilities helps you predict and verify your solutions effectively!

Simplifying Expressions

Simplifying expressions is a crucial step in solving any equation, including quadratic ones. After applying the quadratic formula, you must simplify the resulting expressions to achieve clear and concise results.
In our exercise, the steps took us from \( x = \frac{-4 \pm \sqrt{32}}{2} \) to \( x = -2 \pm 2\sqrt{2} \). Let's break down these simplification steps:

• \( \sqrt{32} \) simplifies to \( 4\sqrt{2} \) because 32 can be factored into 16 and 2, and the square root of 16 is 4.
• The expression \( \frac{-4 \pm 4\sqrt{2}}{2} \) allows dividing each term in the numerator by 2, resulting in \( -2 \pm 2\sqrt{2} \).

By simplifying expressions, you not only make your solutions clearer but also avoid errors. Simplification helps to reveal the most reduced and straightforward forms of the answers, allowing for easier interpretation and verification of your results. Always aim for simplicity to enhance understanding and accuracy.