Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ (x+2)^{2}-x^{2}=4(x+1) $$

Step-by-Step Solution

Answer

The equation is true for all values of x.

Step 1: Apply the formula for the square term

On the left-hand side of the equation, apply the formula for the square term to simplify it. The formula is \((a+b)^2= a^2 + 2ab + b^2\). The equation becomes \(x^2+4x+4 - x^2 = 4(x+1)\).

Step 2: Simplify the equation

Next, simplify the equation by removing like terms and distributing 4 on the right-hand side. This results in: \(4x+4 = 4x+4\).

Step 3: Compare the simplified equation's sides

Upon simplifying, we realize that the left-hand side is equal to the right-hand side. It's always true regardless of the value of x.

Key Concepts

Quadratic Equations

Quadratic equations are an essential concept in algebra, identifiable by their highest degree of the variable being two. The general form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). Solving these equations involves finding values for \(x\) that make the equation true.

Solutions can be found using a variety of methods including factoring, completing the square, using the quadratic formula, or graphing. Depending on the values of \(a\), \(b\), and \(c\), a quadratic equation may have two real solutions, one real solution, or no real solutions when the discriminant (\(\)b^2 - 4ac\(\)) is negative.

Square of a Binomial Formula

The square of a binomial formula is a powerful tool for expanding expressions of the form \( (a+b)^2 \). This formula states that \( (a+b)^2 = a^2 + 2ab + b^2 \), which allows for quick simplification of squared terms without multiplying the binomial by itself.

For example, in the exercise \( (x+2)^2 \), the square of the binomial is applied to become \( x^2 + 4x + 4 \). It's crucial to remember that this formula only applies to squares of binomials, not other exponents, and that the middle term is always twice the product of the two terms in the binomial.

Equation Simplification

Simplifying equations is a key step in solving them. It involves combining like terms, distributing multipliers, and eliminating terms on both sides of the equation. Simplification can sometimes reveal solutions directly or make it easier to apply other solution methods.

In the current exercise, simplification leads to the elimination of the \(x^2\) term and a clearer view of the remaining terms. After distributing and simplifying, you get \(4x + 4 = 4x + 4\), which indicates that all values of \(x\) satisfy the equation, highlighting the importance of simplification in recognizing different types of solutions.

No Solution Equations

Equations that have no solution are also known as inconsistent or contradictory equations. These occur when simplification leads to a statement that is always false, such as \(0 = 3\). In contrast, some equations simplify to a truth, like \(0 = 0\), which implies the equation has an infinite number of solutions, not 'no solution'.

In the given exercise, even though the terms 'no solution equations' were mentioned, the result after simplification indicated that any value for \(x\) would solve the equation, giving us an identity—an equation that is always true, leading to infinite solutions, not an equation with no solutions.