Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$(4 x+1)^{2}=20$$

Step-by-Step Solution

Answer

The solutions are approximately 0.618 and -1.118.

Step 1: Expand the equation

Start by expanding the left side of the equation. This entails squaring the binomial: \((4x+1)^2 = (4x+1)(4x+1) = 16x^2 + 8x + 1\).

Step 2: Set the equation to zero

Subtract 20 from both sides to set the equation to zero: \(16x^2 + 8x + 1 - 20 = 0\), which simplifies to \(16x^2 + 8x - 19 = 0\).

Step 3: Factor or use the quadratic formula

Since factoring is complex here, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Identify \(a = 16\), \(b = 8\), and \(c = -19\).

Step 4: Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\): \(8^2 - 4(16)(-19) = 64 + 1216 = 1280\). Since 1280 is positive, there are two real solutions.

Step 5: Solve for x

Plug the values into the quadratic formula: \[x = \frac{-8 \pm \sqrt{1280}}{32}\].Calculate \(\sqrt{1280} = 8\sqrt{20}\), so:\[x = \frac{-8 \pm 8\sqrt{20}}{32}\].Simplify this to get: \[x = \frac{-2 \pm 2\sqrt{20}}{8}\] or \[x = \frac{-1 \pm \sqrt{20}}{4}\].

Step 6: Identify the exact solutions

The exact solutions are \(x = \frac{-1 + \sqrt{20}}{4}\) and \(x = \frac{-1 - \sqrt{20}}{4}\).

Step 7: Graph the function

Create a graph of the function \(y = (4x + 1)^2 - 20\), and identify the x-intercepts. These intercepts correspond to the solutions found: approximately \(x \approx 0.618\) and \(x \approx -1.118\).

Key Concepts

Quadratic formula

The quadratic formula is a vital tool used to solve any quadratic equation. It is especially handy when factoring the equation directly seems difficult or impossible. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where:

• \(a\), \(b\), and \(c\) are coefficients from the standard form of the quadratic equation \(ax^2+bx+c=0\).

This formula allows us to find the roots or solutions of the equation directly. These solutions can be real or complex, depending on the value under the square root symbol, known as the discriminant. Using the quadratic formula gives both potential solutions in one go with the \(\pm\) symbol, addressing both branches of the quadratic equation.

Discriminant

The discriminant is a specific part of the quadratic formula, represented by \(b^2 - 4ac\). It helps us determine the nature of the roots of the quadratic equation.

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If it is zero, there is exactly one real solution, where both roots are the same.
• If the discriminant is negative, the solutions are not real numbers but complex numbers.

In our exercise example, the discriminant is calculated as 1280, which is positive. This indicates that there are two real solutions for the equation. Understanding the discriminant is crucial as it tells us what type of roots to expect even before solving completely.

Graphical solution

Graphing a quadratic equation provides a visual confirmation of the solutions. The graph of a quadratic is a parabola, and the solutions to the equation are where the parabola intersects the x-axis, known as the x-intercepts.For the equation \(y = (4x + 1)^2 - 20\), the corresponding graph will show us visually where the solutions occur. These points give us the approximate values of the roots: around \(x \approx 0.618\) and \(x \approx -1.118\). Graphing offers a straightforward way to check if the solutions make sense, ensuring that the algebraic process corresponds with the graphical interpretation. It confirms the correctness of the solutions found algebraically by showing them on a coordinate plane.

Exact solutions

Exact solutions refer to the algebraic results obtained when solving a quadratic equation. They often include irrational numbers expressed with radicals, as seen in the exercise:\[x = \frac{-1 + \sqrt{20}}{4}\] and \[x = \frac{-1 - \sqrt{20}}{4}\]These expressions show the precise locations of the roots. While approximations (like from a graph) can give you a sense of where solutions lie, exact solutions provide the complete mathematical representations, ensuring total accuracy.Finding exact solutions is vital for deeper mathematical work as it retains accuracy across various applications. Utilizing the quadratic formula allows for these solutions to be derived, capturing both possible outcomes. Hence, algebra ensures that the exact roots are mathematically sound without reliance solely on visual estimation.