Problem 29 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ x^{2}-4 x-21=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = -3\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -4\), and \(c = -21\). We are going to solve it using the factoring method.

Step 2: Factor the Quadratic

To factor the equation \(x^2 - 4x - 21 = 0\), we need two numbers that multiply to \(-21\) and add to \(-4\). The numbers \(-7\) and \(+3\) satisfy these conditions because \(-7 \times 3 = -21\) and \(-7 + 3 = -4\).

Step 3: Write the Factored Form

Write \(x^2 - 4x - 21 = 0\) as \((x - 7)(x + 3) = 0\). The quadratic is now expressed as a product of two binomials.

Step 4: Apply the Zero Product Property

According to the Zero Product Property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). Apply this to \((x - 7)(x + 3) = 0\).

Step 5: Solve for x

Solve each equation derived from the Zero Product Property:1. Solve \(x - 7 = 0\), giving \(x = 7\).2. Solve \(x + 3 = 0\), giving \(x = -3\).

Key Concepts

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic equation in a way that allows it to be broken down into simpler parts called factors. This process can make it easier to solve the equation. Quadratic equations often follow the standard form:

• \(ax^2 + bx + c = 0\)

Here, the goal of factoring is to express this equation as the product of two smaller binomials. For instance, consider the quadratic equation \(x^2 - 4x - 21 = 0\). To factor it, we need two numbers that multiply to the constant term \(-21\) and add up to the linear coefficient \(-4\).
These numbers are \(-7\) and \(+3\) because:

• \(-7 \times 3 = -21\)
• \(-7 + 3 = -4\)

With these numbers, the equation can be rewritten in its factored form as

• \((x - 7)(x + 3) = 0\)

Factoring helps simplify the equation and sets the stage for applying further techniques to solve it.

Zero Product Property

The zero product property is a fundamental principle in algebra that comes into play once a quadratic equation has been factored. This property states that if the product of two numbers is zero, then at least one of the numbers must be zero. Mathematically, if \((a)(b) = 0\), then:

• \(a = 0\)
• or \(b = 0\)

When applied to a factored quadratic like \((x - 7)(x + 3) = 0\), the zero product property tells us that either:

• \(x - 7 = 0\)
• or \(x + 3 = 0\)

This allows us to split one problem into two simpler equations that can be tackled individually. Thanks to this property, we can solve for the possible values of \(x\) efficiently.

Solving Quadratic Equations

Solving quadratic equations can be done efficiently using a variety of methods, but in our example, we used factoring and the zero product property. After factoring the quadratic equation \(x^2 - 4x - 21 = 0\) into \((x - 7)(x + 3) = 0\), we leveraged the zero product property to break it down into simpler problems.
Solving for \(x\) through each factor, we approach:

• \(x - 7 = 0\) leads to \(x = 7\)
• \(x + 3 = 0\) leads to \(x = -3\)

Thus, the solutions to the quadratic equation are \(x = 7\) and \(x = -3\). These are the values for \(x\) which satisfy the original equation, confirming that by addressing each part individually, we've efficiently found the solution for the entire quadratic equation. Understanding this process reinforces the power of the factoring method and its integration with the zero product property in quickly finding results.