Problem 70 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ x^{2}+9 x+14=0 $$

Step-by-Step Solution

Answer

Answer: The values of x that satisfy the given quadratic equation are \(x=-7\) and \(x=-2\).

Step 1: Identify the quadratic equation

The given quadratic equation is: $$ x^{2}+9x+14=0 $$

Step 2: Factoring the quadratic equation

To factor the quadratic equation, we need to find two numbers that multiply to 14 and add to 9. In this case, those numbers are 7 and 2. So, we can rewrite the equation as: $$ (x+7)(x+2)=0 $$

Step 3: Solve for x

Now, we have two possibilities as either \((x+7)\) or \((x+2)\) must be equal to zero. Solving both equations separately, we get: $$ x+7=0 \Rightarrow x=-7 $$ and $$ x+2=0 \Rightarrow x=-2 $$

Step 4: Write the solution

So, the two solutions for the quadratic equation are: $$ x=-7,\,-2 $$

Key Concepts

Factoring Quadratic Equations

Factoring is a critical method for solving quadratic equations, which are of the form \(ax^2+bx+c=0\). To factor a quadratic equation means to break it down into simpler expressions (called factors) that can be multiplied together to get the original equation.

For the given quadratic equation \(x^2+9x+14=0\), we look for two numbers that add up to \(b\), the coefficient of \(x\), and multiply to \(c\), the constant term. Once you find such a pair of numbers, \(7\) and \(2\) in our case, you can express the quadratic equation as the product of two binomials: \(x+7\) and \(x+2\), corresponding to the equation \( (x+7)(x+2)=0\).

This method of solving is beneficial as it simplifies finding the roots of the equation, which leads us to our next essential concept.

Quadratic Formula

While factoring is a handy technique, not all quadratic equations can be factored easily. That’s where the quadratic formula comes in. The formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) provides a straightforward means to find the roots of any quadratic equation \(ax^2+bx+c=0\).

The beauty of the quadratic formula lies in its universality; it will always work provided the equation has real solutions. The term under the square root, \(b^2-4ac\), known as the discriminant, tells us about the nature of the roots:

• If \(b^2-4ac > 0\), there are two distinct real roots.
• If \(b^2-4ac = 0\), there is exactly one real root (also called a repeated or double root).
• If \(b^2-4ac < 0\), there are no real roots (the solutions are complex).

In our exercise, using the quadratic formula is not necessary due to the simple nature of the quadratic, however, it’s a valuable tool when factoring isn’t a viable option.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) that make the equation true, essentially solving \(ax^2+bx+c=0\). These roots can also be referred to as the solutions, or zeros, of the equation.

In the exercise, we found that \(x+7=0\) or \(x+2=0\), which lead us to the roots \(x=-7\) and \(x=-2\). Each of these roots corresponds to a point where the graph of the quadratic equation crosses the \(x\)-axis. When factoring, the sign before the numbers in each binomial is key; they must be considered when calculating the roots, hence why \(x+7=0\) yields \(x=-7\), not \(7\).

Understanding how to find the roots through factoring, graphical method, or using the quadratic formula, enables students to analyze and interpret the behavior of quadratic functions in various contexts, which is foundational for many advanced topics in mathematics and applied sciences.