Problem 2 - College Algebra with Modeling and Visualization

Question

Exercises \(1-28:\) Solve the quadratic equation. Check your answers for Exercises \(1-12\). $$ x^{2}-9 x+10=-8 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = 6\).

Step 1: Rearrange the equation

First, move all terms to one side of the equation to set it equal to zero. Add 8 to both sides to obtain the equation: \[x^2 - 9x + 10 + 8 = 0\]which simplifies to:\[x^2 - 9x + 18 = 0\]

Step 2: Factor the quadratic equation

To solve the quadratic equation \(x^2 - 9x + 18 = 0\), we need to factor it. We are looking for two numbers whose product is 18 (constant term) and sum is -9 (coefficient of \(x\)). These numbers are -3 and -6. Therefore, the equation factors as:\[(x - 3)(x - 6) = 0\]

Step 3: Solve for the roots

Set each factor equal to zero to find the solutions for \(x\):1. \(x - 3 = 0\)2. \(x - 6 = 0\)Solving these equations gives:1. \(x = 3\)2. \(x = 6\)

Step 4: Verify the solutions

Substitute \(x = 3\) and \(x = 6\) back into the original equation to verify they are correct.- For \(x = 3\):\[3^2 - 9(3) + 10 = -8 \]Calculating gives \(9 - 27 + 10 = -8\), which is true.- For \(x = 6\):\[6^2 - 9(6) + 10 = -8 \]Calculating gives \(36 - 54 + 10 = -8\), which is true.Both solutions are verified.

Key Concepts

Factoring Quadratics

When dealing with quadratic equations such as the form \( ax^2 + bx + c = 0 \), one useful method to find the solution is factoring. Factoring involves rewriting the quadratic equation as a product of two binomials. For the equation in our example, \( x^2 - 9x + 18 = 0 \), we searched for two numbers that multiply to 18 (the constant term) and sum to -9 (the coefficient of \( x \)).

• The numbers -3 and -6 satisfy these conditions because \( (-3) \times (-6) = 18 \) and \( (-3) + (-6) = -9 \).

Thus, the equation \( x^2 - 9x + 18 \) factors into \((x - 3)(x - 6) = 0\). This technique is helpful for quadratic equations featuring simple integers, making the process straightforward and easy to follow.

Solving Equations

Once a quadratic equation is factored into the form \((x - p)(x - q) = 0\), solving it involves finding values of \( x \) that make each factor equal to zero.

• The Zero Product Property states that if a product of two factors equals zero, at least one of the factors must be zero.

In our example, \( x - 3 = 0 \) and \( x - 6 = 0 \) were the two equations derived from the factored form \((x - 3)(x - 6) = 0\).
Solve each equation individually:

• For \( x - 3 = 0 \), add 3 to each side to get \( x = 3 \).
• For \( x - 6 = 0 \), add 6 to each side to get \( x = 6 \).

These are the roots or solutions of the original quadratic equation. Every quadratic equation will have at most two solutions like these.

Verifying Solutions

After finding the solutions to a quadratic equation, it is crucial to verify that they satisfy the original equation. Verification involves substituting the solutions back into the original equation to ensure that the left-hand side equals the right-hand side.
For our example, the original quadratic equation was \( x^2 - 9x + 10 = -8 \).
Let's check each solution:

• Substitute \( x = 3 \): Calculate \( 3^2 - 9(3) + 10 \), which simplifies to \( 9 - 27 + 10 = -8 \). This is true.
• Substitute \( x = 6 \): Calculate \( 6^2 - 9(6) + 10 \), which simplifies to \( 36 - 54 + 10 = -8 \). This is also true.

Both values of \( x \) satisfy the original equation, confirming they are correct solutions. Verifying ensures that the calculations are accurate and provides reassurance of the correctness of the solutions.