Problem 72 - Algebra for College Students

Question

Explain how you would solve the equation \((x+6)(x-4)\) \(=0\) and also how you would solve \((x+6)(x-4)=\) \(-16 .\)

Step-by-Step Solution

Answer

Solutions: \((x+6)(x-4)=0: x=-6, x=4\); \((x+6)(x-4)=-16: x=2, x=-4\).

Step 1: Identify Each Factor in the Equation

The equation given is \((x + 6)(x - 4) = 0\). This is a product of two factors set to zero. For a product to be zero, at least one of the factors must be zero.

Step 2: Apply the Zero Product Property

To solve \((x + 6)(x - 4) = 0\), we apply the zero product property, setting each factor equal to zero: 1. \((x + 6) = 0\) 2. \((x - 4) = 0\).Solving these individually gives the roots of the equation.

Step 3: Solve Each Factor Equation

Solve each of the simpler equations:1. \(x + 6 = 0\) solves to \(x = -6\).2. \(x - 4 = 0\) solves to \(x = 4\). Therefore, the solutions are \(x = -6\) and \(x = 4\).

Step 4: Set the New Equation for Nonzero Product

Now consider the equation \((x + 6)(x - 4) = -16\). This indicates that the product is not zero, so we don't use the zero product property. We will expand and solve it as a quadratic equation.

Step 5: Expand and Rearrange the Equation

Expand the given product: \((x + 6)(x - 4) = x^2 + 2x - 24\). Now set this equal to \(-16\):\(x^2 + 2x - 24 = -16\).Rearrange to form a standard quadratic equation: \(x^2 + 2x - 8 = 0\).

Step 6: Solve the Quadratic Equation Using the Quadratic Formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 1\), \(b = 2\), and \(c = -8\).Calculate the discriminant: \(b^2 - 4ac = 2^2 - 4 \times 1 \times (-8) = 36\).Thus, \(x = \frac{-2 \pm \sqrt{36}}{2}\), resulting in the solutions \(x = 2\) and \(x = -4\).

Key Concepts

Zero Product Property

Understanding the zero product property is fundamental when dealing with quadratic equations, especially when they are expressed in factored form. Imagine this: if you have two numbers multiplying to give zero, one (or both) of those numbers must also be zero. This is the essence of the zero product property.
Applying this principle, for an equation like \((x+6)(x-4) = 0\), you can set each factor to zero:

• \(x + 6 = 0\) leads to \(x = -6\)
• \(x - 4 = 0\) leads to \(x = 4\)

This simple rule helps find the roots of the equation swiftly. It’s powerful because it directly gives the points where the graph of the quadratic equation intersects the x-axis, representing the solutions or roots of the equation.

Quadratic Formula

When a quadratic equation, such as \(ax^2 + bx + c = 0\), cannot be easily factored, we turn to the quadratic formula. It is a universal tool, providing the solution for any quadratic equation.
The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This neat little formula works by plugging in the values of \(a\), \(b\), and \(c\) from your quadratic equation. Let’s see it in action:

• If we have \(x^2 + 2x - 8 = 0\), we identify \(a = 1\), \(b = 2\), \(c = -8\).
• Calculate the discriminant, \(b^2 - 4ac\), which in this case is \(4 - (-32) = 36\).
• The quadratic formula then gives \( x = \frac{-2 \pm 6}{2} \), resulting in solutions \(x = 2\) and \(x = -4\).

The quadratic formula is perfect for solving quadratics that might not be straightforward to factor or even ideal to try and factor.

Factoring

Factoring is breaking down an expression into its multiplicative components. Think of it like breaking down a complex puzzle into smaller pieces. When dealing with quadratic equations, factoring involves rewriting the equation in a form that can be set to zero and solved using the zero product property.
Take the expression \((x+6)(x-4)\). This expression is already factored because it is written as a product of two binomials. The key idea here is to set the equation to zero and apply the zero product property.

• Thing to note is that not all quadratic equations like \(x^2 + 2x - 8 = 0\) factor neatly. Sometimes you might start by trying to factor it, but find it easier to solve using other methods, such as the quadratic formula.
• Practicing factoring helps in identifying patterns, such as recognizing perfect square trinomials or differences of squares.

Factoring is a valuable tool in simplifying expressions and solving equations, but remember, it's one among many methods and sometimes, especially with complex expressions, using it in conjunction with other techniques like the quadratic formula can be beneficial.