Problem 66 - Introductory Algebra for College Students

Question

Solve each equation and check your solutions. $$(x-3)^{2}+2(x-3)-8=0$$

Step-by-Step Solution

Answer

The solutions of the equation \((x-3)^{2}+2(x-3)-8=0\) are \(x = 2 + \sqrt{21}\) and \(x = 2 - \sqrt{21}\).

Step 1 Transposition

First, transpose the problem equation \((x-3)^{2}+2(x-3)-8=0\) to a standard quadratic equation form. The term \((x-3)^{2}\) expands to \(x^2-6x+9\). Therfore, the equation becomes \(x^2-6x+2x-9-8=0\), which simplifies further to \(x^2-4x-17=0\).

Step 2 Applying Quadratic Formula

Solve the quadratic equation by applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Here in the equation \(x^2 - 4x - 17 = 0\), a = 1, b = -4, c=-17. Insert these values into the quadratic formula to get the solutions.

Step 3 Solving Quadratic Formula

After substituting a, b, and c in the formula we get \(x = \frac{-(-4) \pm \sqrt{(-4)^2-4*1*(-17)}}{2*1}\), that simplifies to \(x = \frac{4 \pm \sqrt{16+68}}{2}\), which further simplifies to \(x = 2 \pm \sqrt{21}\). Therefore, the solutions are \(x = 2 + \sqrt{21}\) and \(x = 2 - \sqrt{21}\).

Step 4 Check your solution

You can cross-check your solutions by substituting the found values for x in the original equation and verify that it holds true. Doing this will confirm the accuracy of the solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra for finding the roots of any quadratic equation, typically in the form \(ax^2 + bx + c = 0\). This formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)

Using this formula involves substituting the coefficients \(a\), \(b\), and \(c\) from your equation into the formula. The term \(b^2-4ac\) is called the discriminant, and it determines the nature of the roots:

• A positive discriminant means two real and distinct solutions.
• A zero discriminant means exactly one real solution.
• A negative discriminant indicates two complex solutions.

In our exercise with the equation \(x^2 - 4x - 17 = 0\), we found the solutions to be \(x = 2 + \sqrt{21}\) and \(x = 2 - \sqrt{21}\), utilizing this essential formula.

Equation Solving

Equation solving involves finding values for variables that make an equation true. When working with quadratic equations, our primary goal is to find the values of \(x\) that satisfy \(ax^2 + bx + c = 0\). There are several methods for solving these equations:

• Factoring
• Completing the square
• Using the quadratic formula

In this particular exercise, the quadratic formula is the chosen method due to its reliability and comprehensive nature. This formula can solve any quadratic equation, even when other methods are less suitable. Solving by substitution into this formula gives us the roots, an essential skill in algebra.

Mathematical Transposition

Transposition is the process of rearranging an equation to make it easier to solve. The goal is to write the equation in a form that reveals the variable we are trying to solve for. In our exercise, we had:

• \((x-3)^2+2(x-3)-8=0\)

Transposition involved expanding and rearranging terms:

• Expanding \((x-3)^2\) to \(x^2-6x+9\)
• Simplifying the expression \(x^2 - 4x - 17 = 0\)

This rearrangement allows us to apply the quadratic formula efficiently. Through careful manipulation of terms, transposition transforms a complex-looking equation into a standard quadratic form, simplifying the solution process.

Expanding Equations

Expanding equations is a critical part of solving quadratic problems. It involves distributing and simplifying expressions to a recognizable form. For instance, in the equation \((x-3)^2+2(x-3)-8=0\), the expansion process included:

• Expanding \((x-3)^2\) into \(x^2-6x+9\).
• Combining and simplifying \(2(x-3)\) and constant terms to get \(x^2 - 4x - 17 = 0\).

This converted the equation from a complex expression to a standard quadratic form, making it ready for substitution into the quadratic formula. Expansion is crucial as it simplifies and prepares an equation for further algebraic manipulations, ultimately leading to the solution.