Problem 35 - Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ x^{2}-6 x-7=0 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Equation

Recognize that the given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -6\), and \(c = -7\).

Step 2: Choose Solution Method

Since the equation is quadratic and does not factor easily into integer factors, use the Quadratic Formula to solve it: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) using the values of \(a\), \(b\), and \(c\).\[b^2 - 4ac = (-6)^2 - 4 \times 1 \times (-7) = 36 + 28 = 64\]

Step 4: Apply the Quadratic Formula

Substitute the values into the Quadratic Formula:\[x = \frac{-(-6) \pm \sqrt{64}}{2 \times 1} = \frac{6 \pm 8}{2}\]

Step 5: Simplify the Solution

Solve for \(x\) using the plus and minus in the formula:1. \(x = \frac{6 + 8}{2} = 7\)2. \(x = \frac{6 - 8}{2} = -1\)

Step 6: Write the Final Solution

The solutions to the equation \(x^2 - 6x - 7 = 0\) are \(x = 7\) and \(x = -1\).

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations. Quadratic equations are characterized by having an expression of the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants. The formula used to find the values of \(x\) that satisfy this equation is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The beauty of the quadratic formula is its ability to solve any standard-form quadratic equation, whether it can be factored easily or not. To apply it, simply identify the coefficients \(a\), \(b\), and \(c\) in your equation, and plug them into the formula. The result will provide you with two potential solutions for \(x\): one derived by using the plus sign and the other by using the minus sign in front of the square root.

Discriminant

The discriminant in a quadratic equation is a key component that influences the nature of the solutions. It is the part of the quadratic formula under the square root symbol, expressed as \(b^2 - 4ac\).

• If the discriminant is positive, the equation has two distinct real solutions.
• If it is zero, the equation has exactly one real solution (also known as a repeated or double root).
• If the discriminant is negative, the equation has no real solutions, but two complex solutions.

For the problem \(x^2 - 6x - 7 = 0\), the discriminant is calculated as \(64\), which is a positive number, indicating that the equation has two distinct real solutions.

Factoring

Factoring is another method to solve quadratic equations, often preferred for simpler equations. The idea is to rewrite the quadratic expression as a product of two binomials. For instance, a factored form would look something like \((x - p)(x - q) = 0\). Solving the equation then involves finding the values of \(x\) such that each binomial equals zero, resulting in solutions \(x = p\) and \(x = q\).In the case of \(x^2 - 6x - 7 = 0\), factoring doesn't result in whole numbers easily. Thus, the quadratic formula was chosen over factoring. However, when factoring is straightforward, it can provide a quicker path to the solution than the quadratic formula.

Solving Quadratics

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). The two primary methods for solving quadratics are:

• Quadratic Formula: Useful for all types of quadratic equations, especially when factoring is complex or impossible.
• Factoring: Good for simple quadratics with rational roots that can be easily rewritten as a product of binomials.

Understanding when and how to use these methods effectively is crucial. For example, in our given problem, the choice of using the quadratic formula was driven by the challenge of factoring the equation into integer factors. Solving quadratics often requires analyzing the equation to decide which method will be most efficient.