Problem 99 - Algebra 1

Question

Solve the equation. $$(x+5)(x+6)=0$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=-5, -6\)

Step 1: Apply the Zero Product Property

Since we have \((x+5)(x+6)=0\), we apply the Zero product property. This property states that for real numbers 'a' and 'b'. If \(ab=0\) then \(a=0\) or \(b=0\). So to solve this equation, set each factor equal to zero.

Step 2: Solve for x in the first factor

In the first factor, \(x+5=0\), subtract 5 from both sides to isolate x. This results in \(x=-5\). So, the first solution is \(x=-5\).

Step 3: Solve for x in the second factor

In the second factor, \(x+6=0\), subtract 6 from both sides to isolate x. This gives the solution \(x=-6\). Thus the second solution is \(x=-6\).

Key Concepts

Zero Product Property

Understanding how to use the zero product property is key in solving equations involving products of factors. This property is a fundamental concept in algebra. It tells us that if two numbers multiply to zero, at least one of the numbers must be zero. For example, if you have an equation like

• \((x+5)(x+6) = 0\)

You can apply the zero product property to solve it. The property works by setting each factor separately equal to zero. So, you split it into two individual equations:

• \(x+5 = 0\)
• \(x+6 = 0\)

By doing this, you essentially break down the problem into simpler parts that are easier to manage. This method helps reduce complex products into straightforward equations, making it easier to identify possible solutions for the variable.

Factoring

Factoring is a powerful tool in algebra that involves breaking down expressions into simpler components, or "factors," that can be multiplied to get the original expression. Factoring allows you to transform seemingly complicated expressions into more solvable ones. When

• you factor an equation such as \(x^2 + 11x + 30 = 0\)

You rewrite it as a product of binomials, like

• \((x+5)(x+6) = 0\)

This step is often vital when working with quadratic equations because it transforms the quadratic into a form that easily allows the application of the zero product property.To factor, look for two numbers that multiply to the constant term (here, 30) and add up to the linear coefficient (11 in this case). Once factored, each binomial set equal to zero can be solved simply, revealing the equation’s solutions.

• Factoring involves recognizing and applying patterns, such as the difference of squares or perfect square trinomials, which can streamline finding factors.

Solving Equations

Solving equations is all about finding the value of variables that satisfy the equation. Quadratic equations, such as

• \((x+5)(x+6)=0\)

require specific techniques to solve. After factoring and applying the zero product property, you must isolate the variable to determine its value. In our example:

• To solve \(x+5 = 0\), subtract 5 from both sides: \(x=-5\).
• To solve \(x+6 = 0\), subtract 6 from both sides: \(x=-6\).

Both solutions

• \(x = -5\) and \(x = -6\)

are the values of the variable that satisfy the equation. This process of solving ensures that every possible solution is found. Besides factoring, quadratic equations can also be solved using other methods like completing the square or utilizing the quadratic formula, depending on what is most convenient for the given equation. Each method has its niche, especially when the equation might not factor neatly using simple integers.