Problem 11 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$(x-2)(x+9)=-10$$

Step-by-Step Solution

Answer

The solutions are \(x = -8\) and \(x = 1\).

Step 1: Expand the Equation

First, let's expand the left side of the equation \[(x - 2)(x + 9) = x^2 + 9x - 2x - 18 = x^2 + 7x - 18\]. So the equation becomes:\[x^2 + 7x - 18 = -10\].

Step 2: Move All Terms to One Side of the Equation

To solve the quadratic equation, we need to set it to zero. Add \(10\) to both sides to achieve this:\[x^2 + 7x - 18 + 10 = 0\].Simplifying this, we get:\[x^2 + 7x - 8 = 0\].

Step 3: Factor the Quadratic Equation

Next, we will factor the equation \(x^2 + 7x - 8 = 0\).We need two numbers that multiply to \(-8\) and add to \(7\). These numbers are \(8\) and \(-1\). So, we factor as:\[(x + 8)(x - 1) = 0\].

Step 4: Solve for x

Set each factor equal to zero and solve:1. \(x + 8 = 0\) leads to \(x = -8\).2. \(x - 1 = 0\) leads to \(x = 1\).Thus, the solutions are \(x = -8\) and \(x = 1\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to find the roots or solutions of quadratic equations. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \). The goal when factoring is to express this quadratic as a product of two binomials.

• The first step is to identify two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term) and add to \( b \) (the coefficient of \( x \)).
• In our exercise, the equation \( x^2 + 7x - 8 = 0 \) was factored into \((x + 8)(x - 1) = 0\).
• This means that \( 8 \) and \(-1\) multiply to \(-8\) and add up to \( 7 \).

By factoring, we turn the quadratic equation into a simpler form that can be solved easily by setting each binomial to zero.

Expanding Polynomials

Expanding polynomials is the process of multiplying out expressions to form a standard equation. For example, given two binomials, such as \((x - 2)(x + 9)\), we multiply each term in one binomial with every term in the other.For the given problem:

• Multiply \( x \) by \( x \) to get \( x^2 \).
• Multiply \( x \) by \( 9 \) to get \( 9x \).
• Multiply \(-2\) by \( x \) to get \(-2x \).
• Finally, multiply \(-2\) by \( 9 \) to get \(-18\).

Combine these terms to get: \(x^2 + 9x - 2x - 18\), which simplifies to \(x^2 + 7x - 18\). Expanding is useful as it converts factored forms back into a standard polynomial equation. This step is crucial in preparing the equation for solving.

Solving by Factoring

Solving a quadratic equation by factoring involves expressing the equation in a factored form and then finding solutions by setting each factor to zero. Once the equation is factored, use the zero-product property:

• For any product \( ab = 0 \), either \( a = 0 \) or \( b = 0 \) must hold true.
• In our example, \((x + 8)(x - 1) = 0\) was factored.
• Setting each factor to zero gives \(x + 8 = 0\) and \(x - 1 = 0\).
• Solving these gives \(x = -8\) and \(x = 1\).

The solutions are thus the values of \(x\) that make the original quadratic equation true. Factoring is often the easiest method to find solutions when the equation can be factored neatly.

Setting Equations to Zero

Setting an equation to zero is a pivotal step in solving quadratic equations. This concept, known as "completing the zero", simplifies solving since it allows using properties such as the zero-product rule.

• Start by expanding or rearranging the given equation so all terms are on one side.
• In our exercise, \((x - 2)(x + 9) = -10\) becomes \(x^2 + 7x - 18 = -10\) after expanding.
• Next, add \(10\) to both sides to set the equation to zero: \(x^2 + 7x - 8 = 0\).

This step transforms the equation into a standard quadratic form that can be factored or solved using another method. Setting a quadratic to zero makes it much simpler to apply further algebraic techniques like factoring.