Problem 12 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ (x+1)^{2}+4=29 $$

Step-by-Step Solution

Answer

Question: Solve the equation \((x+1)^2 + 4 = 29\). Answer: The solutions for the given equation are \(x = 4\) and \(x = -6\).

Step 1: Subtract 4 from both sides of the equation

To isolate the \((x+1)^2\) expression, we need to subtract 4 from both sides of the equation. This gives us the following equation: $$(x+1)^2 + 4 - 4 = 29 - 4$$ $$\Rightarrow (x+1)^2 = 25$$

Step 2: Take the square root of both sides

Now, to isolate the \(x+1\) expression, we need to take the square root of both sides of the equation: $$\sqrt{(x+1)^2}=\pm \sqrt{25}$$ $$\Rightarrow x+1 = \pm 5$$

Step 3: Solve for x

Now we can solve for \(x\) in each of the cases: Case 1: \(x + 1 = 5\) $$x + 1 - 1 = 5 - 1$$ $$\Rightarrow x = 4$$ Case 2: \(x + 1 = -5\) $$x + 1 - 1 = -5 - 1$$ $$\Rightarrow x = -6$$ So, the solutions for the equation \((x+1)^2 + 4 = 29\) are \(x = 4\) and \(x = -6\).

Key Concepts

Solving Algebraic Equations

Algebraic equations are mathematical expressions that involve variables, numbers, and operations. The goal is to find the values of the variables that satisfy the equation. Solving algebraic equations is a fundamental skill in mathematics. It is crucial for understanding more complex concepts in algebra, calculus, and other areas of mathematics. To solve an equation means finding all solutions that make the equation true. For instance, if you are given the equation

• \((x+1)^2+4=29\)

The task is to find the value of \(x\) that makes this equation true. Understanding how to manipulate equations through operations like addition, subtraction, multiplication, and division is key to solving them. The main objective is to isolate the variable, meaning you get \(x\) by itself on one side of the equation. This involves a series of reversible operations, ensuring that each step maintains the equality. Once you have isolated the variable, you will have the solution or solutions to your equation.

Quadratic Equations

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(aeq0\). The equation we are dealing with,

• \((x+1)^2+4=29\)

is a transformed form of a quadratic equation. Once you simplify and rearrange the terms, it becomes

• \((x+1)^2 = 25\)

This is still a quadratic equation, just presented slightly differently. To solve quadratic equations, we often use methods like factoring, completing the square, or using the quadratic formula. However, in this case, the equation already is in a perfect square form: \((x+1)^2\). Taking the square root of both sides simplifies solving the equation further. Remember, when you take the square root of both sides, you need to consider both positive and negative roots since

• for example, both \(5\) and \(-5\) satisfy \((x+1)^2 = 25\).

Steps to Solve Quadratic Equations

Breaking down the process of solving quadratic equations can help understand each step and ensure no mistakes are made. Here's how to tackle the example

• \((x+1)^2+4=29\).

Step 1: Isolate the squared term. Begin by eliminating any constants or coefficients from the side of the equation with the squared term. In this example, subtract 4 from both sides:

• \((x+1)^2=25\).

This simplifies the problem, focusing attention on the core quadratic expression.
Step 2: Solve the quadratic expression. Since it is a simple square in the form \((x+1)^2\), take the square root of both sides. Be sure to include both the positive and negative roots:

• \(x+1=\pm5\).

This shows that there are two possible solutions.
Step 3: Solve for the variable. Solve two separate equations:

• \(x+1=5\)
• \(x+1=-5\)

Subtract 1 in both cases to solve for \(x\):

• In the first case, \(x=4\).
• In the second case, \(x=-6\).

This detailed breakdown shows how different steps fit together to reach the solution efficiently.