Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x|=x^{2}+x-24$$

Step-by-Step Solution

Answer

The solutions for the equation \(|x|=x^{2}+x-24\) are \(x = -√{24}\), \(x = 4\), and \(x = -6\).

Step 1: Set Up the Two Equals

Express the absolute value equation as two separate equations: \(x = x^{2}+x-24\) when \(x\) is positive and \(-x = x^{2}+x-24\) when \(x\) is negative.

Step 2: Rearrange Equations to Zero

Rearrange these two equations to equal zero, which makes it in form of a quadratic equation. Hence, \(x^{2}+x-24 - x = 0\) (equation for \(x\) positive) and \(x^{2}+x-24 + x = 0\) (equation for \(x\) negative) become \(x^{2} - 24 = 0\) and \(x^{2} + 2x - 24 = 0\) respectively.

Step 3: Solve for \(x\) Values in Each Equation

Solving the quadratic equations using the quadratic formula, or factoring, gives the roots of the equations. For the equation \(x^{2} - 24 = 0\), factoring will yield \(x = \pm√{24}\). The roots of the second equation \(x^{2} + 2x - 24 = 0\) (which can be factored to \((x-4)(x+6) = 0\)) are \(x = 4\) and \(x = -6\).

Step 4: Check the Solutions

Substitute the found \(x\) values back into the original equation \(|x|=x^{2}+x-24\). The solution \(x = √{24}\) will not hold in the equation, while \(-√{24}\), \(4\), and \(-6\) will.

Key Concepts

Algebraic Solutions

Algebraic solutions refer to the process of finding the values that satisfy an equation using algebraic methods. When tackling problems like \( |x| = x^{2} + x - 24 \), the goal is to identify the specific values of \( x \) that make the equation true.

One of the key strategies is to remove absolute value expressions by setting up two separate equations based on whether \( x \) is positive or negative. This step is essential because the absolute value of a number is always non-negative, so you must consider both the positive and negative possibilities of \( x \) to find all solutions.

Once the absolute value is removed, you can use traditional algebraic techniques to solve the quadratic equations that result. This often includes rearranging terms, factoring, and applying the quadratic formula, each of which sheds light on the potential values of \( x \). After finding the roots or solutions, it is crucial to verify them by plugging them back into the original equation to check their validity.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically taking the form \( ax^{2} + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The absolute value equation from the exercise, when reformulated without the absolute value, converts into two separate quadratic equations.

In our example, we obtained \( x^{2} - 24 = 0 \) and \( x^{2} + 2x - 24 = 0 \). These quadratic equations are fundamental to algebra and occur frequently across various math topics and applications. Solving them involves finding the values of \( x \) that make the equation true, known as the 'roots' of the equation. There are several methods for solving them, such as graphing, using the quadratic formula, completing the square, and factoring. Each method has its own context in which it is most useful.

Factoring Quadratics

Factoring quadratics is a powerful method for solving quadratic equations that involve breaking down the quadratic polynomial into a product of two binomial expressions. This technique leverages the property that if the product of two numbers (or expressions) is zero, at least one of the numbers must be zero.

In the context of the exercise, the equation \( x^{2} + 2x - 24 = 0 \) can be factored into \( (x - 4)(x + 6) = 0 \). When factoring, we look for two numbers that multiply to give the constant term (in this case, -24) and add up to the coefficient of the linear term (in this case, 2). Factoring is often the quickest way to solve a quadratic equation when the polynomial can be easily decomposed into binomials. However, it's not always possible to factor a quadratic, especially if the roots are not rational numbers.

Solving Quadratic Equations

Solving quadratic equations is a critical skill in algebra. After rewriting an absolute value equation as a quadratic, we determine the solutions that satisfy it. When an equation cannot be factored easily, or when the roots are not rational, the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a reliable tool that can find the roots of any quadratic equation.

In the exercise given, we solve \( x^{2} - 24 = 0 \) by taking the square root of 24. This equation is already in a form that's easy to solve without factoring or using the quadratic formula. Yet, these tools are always at our disposal for more complex equations.

Solving quadratics often requires checking the solutions to ensure they are valid, especially when absolute values are involved. Some solutions may not hold when substituted back into the original equation, and verifying solutions helps to confirm their correctness within the given constraints.