Problem 1 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$x^{2}+4 x+3=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -3\) and \(x = -1\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation in the form of \(ax^2 + bx + c = 0\) where \(a=1\), \(b=4\), and \(c=3\). Our goal is to factor this equation.

Step 2: Find Two Numbers that Multiply to c and Add to b

We need to find two numbers that multiply together to give \(c = 3\) and add up to \(b = 4\). These numbers are 3 and 1 since \(3 \times 1 = 3\) and \(3 + 1 = 4\).

Step 3: Write the Equation in Factored Form

Using the numbers identified in Step 2, rewrite the quadratic as \((x + 3)(x + 1) = 0\). This is because \(x^2 + 4x + 3\) factors into \((x + 3)(x + 1)\).

Step 4: Solve for x Using the Zero Product Property

Set each factor equal to zero to find the solutions for \(x\). For \(x + 3 = 0\), subtract 3 from both sides to get \(x = -3\). For \(x + 1 = 0\), subtract 1 from both sides to get \(x = -1\).

Key Concepts

Factoring Techniques

When solving quadratic equations, one of the most helpful methods is factoring. Factoring involves expressing the quadratic equation in a form that is easier to solve. A quadratic equation is usually in the format of \(ax^2 + bx + c = 0\). To factor it, we seek two numbers that multiply to the product of \(a\) and \(c\), and add up to \(b\).
Let's break this down:

• Identify \(a\), \(b\), and \(c\) in the equation.
• Find numbers that multiply to \(a \times c\) and add to \(b\).
• Rewrite the middle term (\(bx\)) using these two numbers.
• Factor pairs or groups to find the factors.

For example, in the equation \(x^2 + 4x + 3 = 0\), find two numbers that multiply to 3 (which is \(a\times c\), with \(a=1\) and \(c=3\)) and add to 4 (which is \(b\)). Here, those numbers are 3 and 1. Rewrite the equation as \((x + 3)(x + 1) = 0\). This process makes solving quadratic equations straightforward.

Zero Product Property

Once you have factored a quadratic equation, you apply the zero product property to solve it. This principle states that if a product of multiple factors equals zero, at least one of the factors must be zero.
To use this property:

• Set each factor in the equation to zero.
• Solve each resulting equation.

Returning to our factored equation, after factoring to \((x + 3)(x + 1) = 0\), we set each factor equal to zero:

• \(x + 3 = 0\)
• \(x + 1 = 0\)

Solving these equations gives us the values of \(x\). The solutions here are \(x = -3\) and \(x = -1\). The beauty of the zero product property is its straightforwardness — once you have the factors, finding zeros is a matter of simple algebraic manipulation.

Solving Quadratic Equations

The process of solving quadratic equations is one of the fundamentals of algebra. It involves finding the values of \(x\) that satisfy the equation. There are several methods to achieve this, including factoring, which we discussed, completing the square, and using the quadratic formula.
When solving by factoring:

• Rearrange the equation to the standard quadratic form \(ax^2 + bx + c = 0\).
• Factor the quadratic expression into two binomials.
• Apply the zero product property to find the solutions.

In our example, \(x^2 + 4x + 3 = 0\), we used the factoring method: found two numbers that multiply to 3 and add to 4, factored the equation as \((x + 3)(x + 1)\), and found the solutions \(x = -3\) and \(x = -1\).
If factoring seems difficult, remember that quadratic equations can also be tackled using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which is a reliable backup for finding solutions.