Problem 10 - Precalculus

Question

Solve each equation. $$(x+2)(x+1)=x^{2}+11$$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Expand the Left Side

First, we need to expand the left-hand side of the equation \((x+2)(x+1)\). This means distributing each term in the first parenthesis with each term in the second parenthesis.Multiply: \[ x(x+1) + 2(x+1) = x^2 + x + 2x + 2 = x^2 + 3x + 2 \]Now, the equation looks like:\[ x^2 + 3x + 2 = x^2 + 11 \]

Step 2: Simplify the Equation

We need to simplify the equation by subtracting \(x^2\) from both sides to eliminate the \(x^2\) term.This gives us:\[ 3x + 2 = 11 \]

Step 3: Solve for x

Now, isolate the variable \(x\) by first subtracting 2 from both sides of the equation.\[ 3x = 11 - 2 \]Simplify:\[ 3x = 9 \]Next, divide both sides of the equation by 3 to solve for \(x\):\[ x = \frac{9}{3} \]Simplify:\[ x = 3 \]

Step 4: Check the Solution

Finally, substitute \(x = 3\) back into the original equation to ensure it's correct.Original equation: \((x+2)(x+1) = x^2 + 11\)Substitute \(x = 3\):\((3+2)(3+1) = 3^2 + 11\)Calculate:\[ 5 \times 4 = 9 + 11 \]\[ 20 = 20 \]Since both sides are equal, \(x = 3\) is indeed the correct solution.

Key Concepts

Factoring

Factoring is a crucial concept in algebra and it often involves decomposing or breaking down an expression into components that, when multiplied together, form the original expression. In our problem, we started with the expression \[(x+2)(x+1)\]. This expression has already been factored because it shows two binomials that, when multiplied, create another expression. Factoring is the reverse of expansion. If you were given \(x^2 + 3x + 2\) and asked to factor it, you would try to express it as a product of two binomials like \((x+2)(x+1)\). This can also be referred to as factoring quadratics when dealing with equations of the form \(ax^2 + bx + c\). Factoring makes it easier to solve because it breaks the problem into manageable parts.

• Use techniques like identifying common factors in terms.
• Remember that the factors of \(c\) must add to give \(b\).
• Practice with simple expressions to build confidence in recognizing patterns.

The key is to identify binomial pairs whose product equals the quadratic expression.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Once we have expanded and simplified the given equation \((x+2)(x+1) = x^2 + 11\) into \(x^2 + 3x + 2 = x^2 + 11\), we've made it simpler by eliminating like terms from both sides.
The key steps in solving this equation involved the following:

• Simplifying by subtracting similar terms from both sides to isolate \(x\).
• Performing arithmetic operations, like subtraction and division, to further restrict the equation down to \(x = 3\).

It's crucial to perform these steps methodically—subtracting or adding as necessary—until you isolate the variable, making sure each manipulation keeps the equation balanced. These operations lead to determining the true value that satisfies the equation.

Algebraic Manipulation

Algebraic manipulation refers to the various techniques used to rearrange and simplify equations. It involves operations such as distributing, combining like terms, and isolating variables. For example, the initial step in solving \((x+2)(x+1) = x^2 + 11\), was expanding using the distributive property: \[x^2 + x + 2x + 2 = x^2 + 3x + 2\].
Combining like terms is a basic yet vital step in simplifying expressions. After expanding, merging \(x\) and \(2x\) yields \(3x\), simplifying the expression to a more manageable form. Adding, subtracting, multiplying, dividing, and factoring fall under algebraic manipulation, always ensuring each side of the equation stays balanced.

• Simplify expressions step-by-step, focusing on one type of manipulation at a time.
• Consistently check your work against the original equation to preserve equality.
• Remember that each action taken on one side must be mirrored on the other side to maintain balance.

Mastering these basic maneuvers allows deeper exploration into complex algebraic problems.