Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by factoring. $$ x^{2}-9 x+18=0 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = 3\) and \(x = 6\).

Step 1: Identify the Equation Type

Recognize that the given equation \(x^2 - 9x + 18 = 0\) is a quadratic equation in the standard form \(ax^2 + bx + c = 0\), with \(a = 1\), \(b = -9\), and \(c = 18\).

Step 2: Look for Factor Pairs

To factor the equation, look for two numbers that multiply to \(c = 18\) and add to \(b = -9\). These numbers are \(-3\) and \(-6\) since \((-3) \times (-6) = 18\) and \((-3) + (-6) = -9\).

Step 3: Write the Factored Form

Rewrite the quadratic equation using the factor pairs identified. The equation factors to \((x - 3)(x - 6) = 0\).

Step 4: Apply Zero-Product Property

Set each factor equal to zero and solve for \(x\). First, solve \(x - 3 = 0\), which gives \(x = 3\). Then solve \(x - 6 = 0\), which gives \(x = 6\).

Step 5: Verify the Solution

Check the solutions by substituting back into the original equation. For \(x = 3\), \(3^2 - 9(3) + 18 = 0\) holds true. Similarly, for \(x = 6\), \(6^2 - 9(6) + 18 = 0\) holds true.

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations by expressing them as a product of two linear equations. The goal is to rewrite the quadratic equation \(ax^2 + bx + c = 0\) in its factored form \((px + q)(rx + s) = 0\). When dealing with a quadratic that doesn't have a leading coefficient \(a\) greater than 1, this process is more straightforward.
To factor the given quadratic \(x^2 - 9x + 18 = 0\), we need to find two numbers that multiply to the constant term, \(c = 18\), and add to the linear coefficient \(b = -9\).
In this example:

• The two numbers are \(-3\) and \(-6\), since \((-3) \times (-6) = 18\) and \((-3) + (-6) = -9\).
• Using these numbers, we can express the quadratic equation as \((x - 3)(x - 6) = 0\).

This step is crucial as it simplifies solving the equation further with basic algebraic rules.

Zero-Product Property

The zero-product property is a fundamental principle used in algebra to solve equations that have been factored. It states that if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, for any real numbers \(a\) and \(b\), if \(ab = 0\), then \(a = 0\) or \(b = 0\).
This property is essential when solving an equation that has been factored, like \((x - 3)(x - 6) = 0\).
Applying the zero-product property, we set each factor equal to zero:

• \(x - 3 = 0\) which leads to the solution \(x = 3\).
• \(x - 6 = 0\) which leads to the solution \(x = 6\).

By considering each factor separately, we can find the values of \(x\) that satisfy the original quadratic equation.

Solving Equations

Solving equations involves finding the values of the variable that make the equation true. In the context of quadratic equations, once we have factored the equation and applied the zero-product property, solving each of the resulting equations is straightforward.
For the factored equation \((x - 3)(x - 6) = 0\), solving each part:

• The equation \(x - 3 = 0\) produces the solution \(x = 3\).
• The equation \(x - 6 = 0\) produces the solution \(x = 6\).

These solutions tell us the values of \(x\) that make the quadratic equation \(x^2 - 9x + 18 = 0\) true. Solving quadratic equations through this method can be particularly efficient, especially when the quadratic is easily factorable.

Verification of Solutions

Verification of solutions is a critical step to ensure that the obtained answers are correct. After solving a quadratic equation by factoring, each solution should be substituted back into the original equation to check its validity.
For our solutions \(x = 3\) and \(x = 6\):

• Substitute \(x = 3\) into the original equation: \(3^2 - 9(3) + 18 = 0\), which simplifies to \(9 - 27 + 18 = 0\), validating that \(3\) is a correct solution.
• Substitute \(x = 6\) into the original equation: \(6^2 - 9(6) + 18 = 0\), which simplifies to \(36 - 54 + 18 = 0\), verifying that \(6\) is also a correct solution.

This step confirms the accuracy of the solutions and reinforces an understanding of how the solutions satisfy the original quadratic equation.