Problem 37 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$x^{2}-49=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(x^{2}-49=0\) are \(x=-7\) and \(x=7\)

Step 1: Factorize the equation

Using the difference of squares factoring rule, which states that \(a^{2}-b^{2} = (a+b)(a-b)\), factorize the quadratic equation to be: \((x+7)(x-7)=0\)

Step 2: Find the roots

Set each factor equal to zero and solve for \(x\). This gives the roots of the equation as \(x=-7\) and \(x=7\)

Step 3: Check the roots

Substitute these values back into the original equation. If the equation is true, then the roots are correct. For \(x=-7\), we have \((-7)^{2}-49=0\) which is true, and for \(x=7\), we have \((7)^{2}-49=0\) which is also true. Therefore, the roots are correct.

Key Concepts

Difference of Squares

Factoring a quadratic equation can seem like a tricky process, but recognizing certain patterns can make it simpler. One such pattern is the 'difference of squares,' which describes a scenario where the equation takes the form \(a^2 - b^2\). Given this pattern, the equation can be neatly factored into \(a + b)(a - b)\). To picture this with real numbers, let's look at the equation \(x^2 - 49 = 0\). Here, we can identify \(a = x\) and \(b = 7\), since \(7^2 = 49\). Subsequently, applying the difference of squares rule yields \(x + 7)(x - 7) = 0\). This factoring pattern is especially convenient because it's a quick step that doesn't involve complex calculations, making it a favorite shortcut among students tackling quadratic equations.

Finding Roots

Once we factor the quadratic equation, the next step is to find its roots, or the values of \(x\) that satisfy the equation \(x^2 - 49 = 0\). After the equation has been factored into the form \(x + 7)(x - 7) = 0\), determining the roots is straightforward.

To find these roots, we set each factor equal to zero and solve for \(x\). So, we get two equations: \(x + 7 = 0\) and \(x - 7 = 0\). Solving these, we find \(x = -7\) and \(x = 7\). These solutions are the points where the graph of the quadratic equation crosses the \(x\)-axis. In more complex cases, finding roots might require additional techniques, such as completing the square or using the quadratic formula, but in the case of difference of squares, the process is much more efficient.

Quadratic Equation Check

It's crucial to verify that the roots we find are correct. This step, often termed as the 'quadratic equation check,' involves substituting the roots back into the original equation to confirm that they produce true statements.

For instance, with the roots \(x = -7\) and \(x = 7\) for the equation \(x^2 - 49 = 0\), we substitute each value for \(x\) and check the validity. With \(x = -7\), we get \( (-7)^2 - 49 = 0\), which simplifies to \(49 - 49 = 0\), that is indeed true. Similarly, substituting \(x = 7\) leads to \( (7)^2 - 49 = 0\), which also confirms \(49 - 49 = 0\). If at any point the substitution does not result in a true statement, it indicates a potential mistake in the process, prompting us to review and correct our work. Performing such checks ensures the accuracy of our answers and solidifies our understanding of the concepts at play.