Problem 58 - Algebra 2

Question

CHALLENGE. Explain how you would solve \(|a-3|^{2}-9|a-3|=-8 .\) Then solve the equation.

Step-by-Step Solution

Answer

The solutions are \(a = 11, -5, 4,\) and \(2\).

Step 1: Introduce a New Variable

Let us define a new variable \(u = |a-3|\). This simplifies our original equation into a quadratic equation: \(u^2 - 9u = -8\).

Step 2: Rewrite the Equation

Rearrange the equation \(u^2 - 9u = -8\) to standard quadratic form: \(u^2 - 9u + 8 = 0\).

Step 3: Factor the Quadratic Equation

Factor the quadratic equation \(u^2 - 9u + 8 = 0\). Notice that the factors of 8 that add up to 9 are 8 and 1. Thus, the equation factors as \((u - 8)(u - 1) = 0\).

Step 4: Solve for u

Set each factor equal to zero: \(u - 8 = 0\) and \(u - 1 = 0\). This gives us two potential solutions for \(u\): \(u = 8\) and \(u = 1\).

Step 5: Back-Substitute to Determine a

Recall that \(u = |a-3|\). For \(u = 8\), \(|a-3| = 8\), giving the two possible equations: \(a - 3 = 8\) and \(a - 3 = -8\). Solving these gives \(a = 11\) and \(a = -5\), respectively. For \(u = 1\), \(|a-3| = 1\), giving \(a - 3 = 1\) and \(a - 3 = -1\), resulting in \(a = 4\) and \(a = 2\).

Step 6: Verify Solutions

Substitute \(a = 11, -5, 4,\) and \(2\) back into the original equation \(|a-3|^{2}-9|a-3|=-8\) to verify that each satisfies the equation. All four values satisfy the original equation, confirming them as valid solutions.

Key Concepts

Quadratic Equations

Quadratic equations represent expressions that involve at most the square of a variable. These equations generally take a standard form given by \(ax^2 + bx + c = 0\). Understanding the structure of a quadratic is crucial, as it provides the framework needed to solve the equation. In our example, the transformed quadratic equation is \(u^2 - 9u + 8 = 0\). The coefficients correspond to \(a = 1\), \(b = -9\), and \(c = 8\). Recognizing these components helps in applying methods like factoring or using the quadratic formula for solutions. Key characteristics of quadratic equations include:

• The presence of a squared term \(u^2\).
• A linear term \(-9u\).
• A constant term \(8\).

Quadratic equations often result in two solutions because the equation forms a parabola in a graph setting, crossing the x-axis at these solutions.

Factoring

One effective method to solve quadratic equations is factoring. This technique involves expressing the quadratic as a product of its factors. When you successfully break down the equation into factors, it allows you to set each factor equal to zero to find the solutions.Let's take our equation \(u^2 - 9u + 8 = 0\). To factor it, we look for two numbers that multiply to 8 (the constant term) and add to -9 (the coefficient of the linear term). These numbers are -1 and -8. Consequently, the factored form of the equation is \((u - 8)(u - 1) = 0\).The benefit of factoring is that it turns a complex quadratic equation into a simple algebraic expression, making it easier to solve. This method gives us the solutions by setting each factor equal to zero, producing the solutions for \(u\). Therefore, \(u = 8\) and \(u = 1\).

Variable Substitution

Variable substitution simplifies complex equations by introducing a new variable. This step is pivotal, especially when simplifying expressions involving absolute values or non-linear terms.In this exercise, we substitute \(|a-3|\) with \(u\) to transform the original problem \(|a-3|^{2} - 9|a-3| = -8\) into a more manageable quadratic equation \(u^2 - 9u + 8 = 0\). This conversion helps in streamlining our approach to focus on solving a quadratic equation instead of wrestling with absolute value terms directly.By solving the substituted equation first, we find values for \(u\). Then, substituting back \(u = |a-3|\), we get \(|a-3| = 8\) or \(|a-3| = 1\). These equations, when further simplified, provide possible values for \(a\) by solving two separate simple linear equations for each value of \(u\).

Solution Verification

Verifying a solution involves confirming that each potential solution satisfies the original equation. This step ensures that no errors were made in the transformation or calculation process.Once the values for \(a\) are found through back substitution, \(a = 11, -5, 4,\) and \(2\), we substitute these back into the original expression \(|a-3|^2 - 9|a-3| = -8\) to check correctness. This substitution confirms whether these values truly uphold the initial equation.This validation step is crucial because transformations like variable substitution can sometimes lead to extraneous solutions. By plugging the values back, we ascertain that each one satisfies the condition set by the original equation, confirming the thoroughness and accuracy of the solution.