Problem 9 - Algebra and Trigonometry

Question

Solve equation by factoring. $$ 3 x^{2}+12 x=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(3x^{2}+12x=0\) are \(x=0,-4\).

Step 1: Simplify the Equation

The equation is \(3x^{2}+12x=0\). We can factor out a \(3x\) to simplify it, giving us: \(3x(x+4)=0\)

Step 2: Set Each Factor Equal to Zero

Now, each factor of the equation can be set equal to zero to solve for \(x\) respectively. That'll give us two equations: \(3x=0\) and \(x+4=0\)

Step 3: Solve Each Equation

Now if we solve these two equations, for \(3x=0\), we get \(x=0\). While for \(x+4=0\), if we subtract 4 from both sides, we get \(x=-4\)

Key Concepts

Factoring Quadratics

When solving quadratic equations like \(3x^2 + 12x = 0\), the first step is usually to simplify the equation through factoring. Factoring transforms the quadratic into a product of two or more simpler expressions. Think of factoring quadratics as finding two numbers that multiply to give you the original coefficient of \(x^2\) (here it's 3) and adding to the coefficient of \(x\) (here it's 12).

In our example, we can factor out the greatest common factor (GCF), which is \(3x\) in this case, resulting in: \[3x(x + 4) = 0\]. Once factored, it becomes easier to identify the solutions through the zero product property, which we will discuss in the next section.

Zero Product Property

The zero product property is a fundamental principle in algebra stating that if the product of two factors is zero, then at least one of the factors must be zero. For our equation, \(3x(x + 4) = 0\), the product is zero, implying that either \(3x = 0\) or \(x+4 = 0\) must be true.

By setting each factor equal to zero, as done in step 2 of our exercise, we're applying this property. It's a powerful tool because it breaks down a complex equation into simpler ones that are easier to solve, leading us closer to the algebraic solutions, the topic of our next concept.

Algebraic Solutions

Algebraic solutions are the values of the variable that satisfy the original equation. In our exercise, after applying the zero product property, we have two smaller equations: \(3x = 0\) and \(x + 4 = 0\). Solving these gives us the algebraic solutions to the quadratic equation.

For \(3x = 0\), dividing both sides by 3 results in \(x = 0\). Similarly, for \(x + 4 = 0\), subtracting 4 from both sides yields \(x = -4\). These values, 0 and -4, are the roots of the original quadratic equation.

Simplifying Equations

The process of simplifying equations makes them easier to solve. This can involve factoring, as we saw with the quadratic, or carrying out operations to isolate the variable. When we simplify, we're trying to strip the equation down to its most basic form, where the solution becomes apparent.

In the given exercise, factoring out the common factor was the first step of simplification. Then, we used basic arithmetic operations (such as division and subtraction) to further simplify and solve the equations. These techniques are essential for students to learn and apply, as they convert seemingly complex problems into straightforward ones.