Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by factoring. $$ x^{2}+4 x=12 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x=2\) and \(x=-6\).

Step 1: Rearrange the equation

Rearrange the equation into standard quadratic format by moving all terms to one side of the equation. This can be achieved by subtracting 12 from both sides to keep the equation balanced, giving us \(x^{2}+4 x-12=0\).

Step 2: Factor the quadratic equation

Now factor the quadratic equation. Look for two numbers that add up to 4 (the coefficient of x) and multiply to -12 (the constant term). The numbers that satisfy this are 6 and -2, so we can write the equation as \((x-2)(x+6) =0\).

Step 3: Solve for x

Each factor set to zero gives a solution for x. Therefore, set \(x-2 = 0\) and \(x+6 = 0\). Solving these gives \(x=2\) and \(x=-6\) as the solutions to the equation.

Key Concepts

Factoring

To solve a quadratic equation through factoring, the goal is to express the equation in terms of products, using its roots or zeros. Factoring involves breaking down the equation into its simplest parts. In the exercise, the equation is given as \(x^2 + 4x = 12\). The first step in factoring is to place the equation into the standard quadratic form, but more on that later.

The heart of factoring is to identify two numbers that satisfy two conditions:

• They add up to the coefficient of the linear term (in this case, the number next to \(x\), which is 4).
• They multiply to the constant term when the equation is rearranged, which becomes -12.

By finding suitable numbers, which are 6 and -2 for this problem, we can write the factored form of the equation as \((x-2)(x+6) = 0\).

Standard Quadratic Form

The standard quadratic form of an equation is essential as it sets the groundwork for effective factoring or using other methods like the quadratic formula. The standard quadratic form is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable.

When we have an equation like \(x^2 + 4x = 12\), the first step is to rearrange it so that it follows the standard form. This involves moving all terms to one side, typically the left, to set the equation equal to 0. By subtracting 12 from both sides, the equation becomes \(x^2 + 4x - 12 = 0\). This rearrangement helps in applying further techniques such as factoring or using the quadratic formula. By setting it this way, it becomes much easier to identify potential factors or apply other methods effectively.

Solving Equations

The ultimate aim of factoring and rearranging into the standard quadratic form is to solve the quadratic equation effectively. Once the equation like \((x-2)(x+6) = 0\) is factored, solving it becomes straightforward.

The zero-product property is a key concept here, which states that if a product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor to zero and solve for \(x\):

• For \(x-2 = 0\), add 2 to both sides to get \(x = 2\).
• For \(x+6 = 0\), subtract 6 from both sides to get \(x = -6\).

Both values represent the solutions to the quadratic equation. These solutions, \(x = 2\) and \(x = -6\), are where the original quadratic equation equals zero. By successfully carrying out these steps, you've efficiently used factoring to solve the quadratic equation.