Problem 6 - Algebra and Trigonometry

Question

Solve the equation by factoring. $$ x^{2}+3 x=4 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -4\).

Step 1: Convert Equation to Standard Form

First, we need to move all terms to one side of the equation to set it to equal 0. Starting with the given equation: \[ x^2 + 3x = 4 \]Subtract 4 from both sides:\[ x^2 + 3x - 4 = 0 \]

Step 2: Identify a and c for Splitting Middle Term

Now that the equation is in the form \( ax^2 + bx + c = 0 \), identify \( a = 1 \), \( b = 3 \) and \( c = -4 \). We need two numbers that multiply to \( a \cdot c = -4 \) and add up to \( b = 3 \).

Step 3: Find Numbers for Splitting Middle Term

The numbers that multiply to \(-4\) and add to \(3\) are \(4\) and \(-1\) because \(4 imes (-1) = -4\) and \(4 + (-1) = 3\).

Step 4: Rewrite the Equation with Split Middle Term

Use the numbers found to break up the middle term:\[ x^2 + 4x - x - 4 = 0 \]Now the terms are split so we can factor by grouping.

Step 5: Factor by Grouping

Group the terms and factor each group:\[ (x^2 + 4x) + (-x - 4) = 0 \]Factor out the greatest common factor from each group:\[ x(x + 4) - 1(x + 4) = 0 \]

Step 6: Factor Out the Common Binomial Factor

Notice \((x + 4)\) is a common factor:\[ (x - 1)(x + 4) = 0 \]Set each factor equal to zero to find the roots.

Step 7: Solve for the Roots

Set each factor equal to zero and solve for \( x \):\(x - 1 = 0\) or \(x + 4 = 0\).Solving these gives:\(x = 1\) or \(x = -4\).

Key Concepts

Factoring

Factoring is a fundamental algebraic skill used to solve quadratic equations. The main purpose of factoring is to express a polynomial as a product of its simpler polynomials. To solve quadratics by factoring, we convert the equation into a standard form and look for a way to break it down into two binomials.
Let's explore this with an example: Suppose we have a quadratic equation like \( x^2 + 3x - 4 = 0 \). Our goal is to factor it.

• First, we look for two numbers that multiply to the product of the coefficient of \( x^2 \) (which is 1) and the constant term (-4), meaning we need numbers that multiply to -4.
• Additionally, these two numbers must add up to the coefficient of \( x \) (which is 3).
• In this case, the numbers 4 and -1 meet these requirements because \( 4 imes -1 = -4 \) and \( 4 + (-1) = 3 \).

Once we identify these numbers, we rewrite the quadratic equation by splitting the middle term using them. The equation \( x^2 + 3x - 4 = 0 \) becomes \( x^2 + 4x - x - 4 = 0 \). This allows us to group and factor each part, eventually leading us to the solution.

Polynomial Roots

Understanding polynomial roots is key to solving quadratic equations. Polynomial roots are the solutions, values of \( x \), that make the equation true (i.e., make the polynomial equal zero). For a quadratic equation like \((x - 1)(x + 4) = 0\), the roots are found where each factor equals zero.
Consider the equation after factoring: \((x - 1)(x + 4) = 0\).

• We set each factor to zero: \( x - 1 = 0 \) and \( x + 4 = 0 \).
• Solving these simple equations gives the values \( x = 1 \) and \( x = -4 \).

This means when \( x \) is either 1 or -4, the entire polynomial evaluates to zero, confirming these values as the roots. Understanding roots helps us see where the graph of a quadratic intersects the x-axis. Two distinct roots indicate two intersection points, which is what we see here.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying polynomial expressions to solve equations effectively. In our problem, you start by converting the original equation \( x^2 + 3x = 4 \) into standard form \( x^2 + 3x - 4 = 0 \). This step is crucial as it sets the stage for factoring.
Upon reaching standard form, we focused on rearranging terms through techniques such as splitting the middle term and grouping. Let's walk through the manipulation steps:

• Recognize that the equation must equal zero to allow for proper factorization.
• Identify the coefficients and constant term to find possible pairings of numbers for the middle term split.
• Utilize these pairings to rewrite and group terms effectively for factorization.

Mastering algebraic manipulation makes solving complex equations more straightforward. By honing these skills, you improve your ability to break down complex polynomials into simpler forms, thus providing easier paths to solutions.