Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$(x+5)^{2}+3=(x-2)^{2}$$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{12}{7}\).

Step 1: Expand Both Sides

First, we need to expand each side of the equation. The left side gives us: \((x+5)^{2} = x^2 + 10x + 25\)The right side gives us: \((x-2)^{2} = x^2 - 4x + 4\).This transforms the equation into: \(x^2 + 10x + 25 + 3 = x^2 - 4x + 4\)

Step 2: Simplify the Equation

Combine like terms on the left side of the equation: \(x^2 + 10x + 28 = x^2 - 4x + 4\).Subtract \(x^2\) from both sides to eliminate the quadratic term:\(10x + 28 = -4x + 4\).

Step 3: Rearrange Terms

Add \(4x\) to both sides to combine the x terms:\(10x + 4x + 28 = 4\)This simplifies to:\(14x + 28 = 4\).

Step 4: Isolate x

Subtract 28 from both sides to isolate the terms with x:\(14x = 4 - 28\)This simplifies to:\(14x = -24\).

Step 5: Solve for x

Divide both sides by 14 to solve for x:\(x = \frac{-24}{14} = \frac{-12}{7}\).So the solution is:\(x = -\frac{12}{7}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to solve. It's crucial in solving quadratic equations because it allows us to isolate the variable we need to find. In this particular problem, algebraic manipulation helps us simplify both sides of the given equation after expanding them.
Here’s how it's done in the step-by-step solution:

• Once both sides of the equation are expanded, notice the equation contains the quadratic term \(x^2\) on both sides. By subtracting \(x^2\) from both sides, the quadratic terms are eliminated, simplifying the equation to a linear form.
• This reduction allows for the rest of the equation to be simplified using basic arithmetic operations, keeping manipulating steps straightforward and clear until we isolate the variable \(x\).

Algebraic manipulation often requires performing several small operations in sequence, like addition, subtraction, multiplication, or division, to maintain an equation's balance while simplifying it.

Expanding Expressions

Expanding expressions involves multiplying out the terms in an equation so that it becomes easier to manage and solve. In the given problem, both sides of the equation involve squaring binomials.
Here’s the process step by step:

• To expand \((x + 5)^2\), you use the formula \((a + b)^2 = a^2 + 2ab + b^2\), which results in \(x^2 + 10x + 25\).
• Similarly, expand \((x - 2)^2\) using the same formula where \(a = x\) and \(b = -2\). This expansion gives \(x^2 - 4x + 4\).

After expansion, the problem simplifies to manageable polynomial equations, setting the stage for further simplification and, ultimately, solving for \(x\). The key benefit of expanding expressions is that it transforms the equation into a format where algebraic manipulation can be used effectively.

Solving Linear Equations

Once the equation is simplified to a linear form, the next step is solving the linear equation. Linear equations have the general form \(ax + b = c\), where the solution is straightforward as it involves isolating the variable \(x\).
From the simplified form \(14x + 28 = 4\):

• First, subtract 28 from both sides to move the constant terms to one side of the equation, resulting in \(14x = -24\).
• Second, divide both sides by 14 to isolate \(x\). This gives \(x = \frac{-24}{14}\), which can be further simplified to \(x = -\frac{12}{7}\).

By following these steps, you arrive at the solution for \(x\). Solving linear equations is a fundamental skill in algebra, as it frequently forms the basis for finding solutions in more complex algebraic expressions.