Problem 6 - 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 \(x^{2}-6 x=7\).

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = -1\).

Step 1: Rewrite the Equation

First, move all terms to one side of the equation to set it equal to zero. Start with the equation \(x^2 - 6x = 7\) and subtract 7 from both sides:\[x^2 - 6x - 7 = 0\]

Step 2: Identify Coefficients for Quadratic Formula

Now that the equation is in standard quadratic form \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 1\), \(b = -6\), \(c = -7\).

Step 3: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions. Substitute the values into the formula:\[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1}\]

Step 4: Simplify the Expression

Calculate the discriminant and simplify the expression:- First calculate the discriminant: \[b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-7) = 36 + 28 = 64\]- Substitute back and find x:\[x = \frac{6 \pm \sqrt{64}}{2}\]

Step 5: Solve for x

Now compute the two possible solutions:- Since \(\sqrt{64} = 8\), we have:\[x = \frac{6 + 8}{2} = 7\]\[x = \frac{6 - 8}{2} = -1\]

Step 6: Verify the Solutions

Verify both solutions by substituting back into the original equation:- For \(x = 7\):\[7^2 - 6 \times 7 = 49 - 42 = 7 \text{ (True)}\]- For \(x = -1\):\[(-1)^2 - 6 \times (-1) = 1 + 6 = 7 \text{ (True)}\]

Key Concepts

Quadratic Formula

When you encounter a quadratic equation like the one given in the problem, a powerful tool at your disposal is the quadratic formula. This formula simplifies the process of finding solutions to any quadratic equation and is applicable in almost every situation. The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For the formula to work, your quadratic equation must be in the standard form. When using this formula, here's what each term means:

• a, b, and c are the coefficients from the standard form of the quadratic equation \(ax^2 + bx + c = 0\).
• -b quickly changes the sign of the coefficient \(b\).
• \(\sqrt{b^2 - 4ac}\) is the square root of the discriminant of the equation. This part determines the nature of the solutions, whether they are real or complex.
• The \(\pm\) symbol indicates that there will usually be two solutions: one by adding and one by subtracting.

The formula elegantly provides you the roots based on these calculations, and it includes every possible outcome for any quadratic equation.

Standard Form of a Quadratic Equation

Before applying the quadratic formula, it’s crucial to understand the importance of the standard form. A quadratic equation must first be converted into a specific arrangement known as the standard form \[ax^2 + bx + c = 0\]Here's how you break it down:

• The term \(ax^2\) is the quadratic term, where \(a\) is the coefficient of \(x^2\).
• The term \(bx\) represents the linear term, where \(b\) is the coefficient of \(x\).
• c is the constant term without any variable attached to it.

To convert any quadratic equation into its standard form, all terms must be combined on one side of the equation with zero on the other side. In your exercise, you started with the equation \(x^2 - 6x = 7\). By subtracting 7 from both sides, you converted it to the standard form \(x^2 - 6x - 7 = 0\). This arrangement is critical for identifying the coefficients that will be used in the quadratic formula.

Discriminant

The discriminant is a key component found under the square root in the quadratic formula. It provides valuable information about the nature of the roots or solutions of the quadratic equation. The discriminant is defined as:\[b^2 - 4ac\]Understanding what this value signifies can help you predict the type of solutions you might expect:

• If \(b^2 - 4ac > 0\), the equation has two distinct real solutions. A positive discriminant means the graph of the equation intersects the x-axis in two places.
• If \(b^2 - 4ac = 0\), there's exactly one real solution, or a double root. The graph touches the x-axis but doesn’t cross it.
• If \(b^2 - 4ac < 0\), the equation has no real solutions, but two complex solutions. Here, the graph does not intersect the x-axis at all.

In the exercise you worked through, the discriminant calculated was 64. Since 64 is greater than zero, it indicates two distinct real solutions, confirming the roots you found: \(x = 7\) and \(x = -1\). Thus, understanding the discriminant allows you to anticipate the nature of the solutions you will find with the quadratic formula.