Problem 40 - 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. $$9 x^{2}-100=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(9x^2 -100 = 0\) are \(x = -10/3\) and \(x=10/3\).

Step 1: Rearrange the equation

The given equation is \(9x^2 - 100 =0\), but factoring will be easier if this equation is arranged in the format \(ax^2 + bx + c = 0\). So, rewrite the equation as \(9x^2 = 100\).

Step 2: Factor the equation

Factoring the equation actually means expressing the equation in a multiplicative form. In this step, the right side of the equation which is \(100\) can be rewritten as \((10)^2\) and here \(9x^2\) as \((3x)^2\). The equation can now be factored following the difference of squares method: \(a^2 - b^2 = (a - b)(a + b)\). Applying this to our equation, we have \((3x + 10)(3x - 10) = 0\).

Step 3: Solve for x

Next step is to set each factor equal to zero and then solve for \(x\). So, from \((3x + 10) = 0\), solving for \(x\) gives \(x = -10/3\). And from \((3x - 10) = 0\), solving for \(x\) gives \(x = 10/3\).

Step 4: Verify the solution

The final step is to verify the solution by substituting the obtained \(x\) values back into the original equation. Substituting \(x = -10/3\) and \(x=10/3\) into the original equation should result in a true statement. When checking, keep in mind that both the left side and right side of the equation must be equal. After checking you'll find that they indeed produce \(0=0\), thereby confirming the solution as correct.

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is about finding the values of \(x\) that satisfy the equation. A quadratic equation is usually in the form \(ax^2 + bx + c = 0\). The goal is to find the values of \(x\) that make this statement true. Quadratic equations can be solved by various methods including factoring, completing the square, using the quadratic formula, or graphing.

Factoring especially comes in handy when the quadratic can be quickly rewritten in a simpler product form. For our equation, \(9x^2 - 100 = 0\), the factoring method requires finding two terms that multiply to the original quadratic. Once factoring is done, we set each factor equal to zero and solve for \(x\). This gives us the solutions: \(x = -\frac{10}{3}\) and \(x = \frac{10}{3}\).

Remember, solving quadratic equations by factoring is one of the quickest methods if the equation can be easily expressed as a product of binomials.

Difference of Squares

The difference of squares is a special pattern used to simplify and solve equations like the one in our exercise: \(9x^2 - 100 = 0\). It takes the form \(a^2 - b^2\) where \(a\) and \(b\) are any expressions.The rule is:

• \(a^2 - b^2 = (a + b)(a - b)\)

This simplification is very useful because it transforms a complex equation into something much simpler.

When we take the equation \(9x^2 - 100 = 0\), we can write it as \((3x)^2 - (10)^2 = 0\). According to the difference of squares rule, this becomes \((3x + 10)(3x - 10) = 0\). Solving this, by setting each parenthesis to zero, we find the solutions for \(x\).

Always be on the lookout for equations that fit the difference of squares pattern. It's a handy shortcut that simplifies your work considerably.

Verification of Solutions

Once you have potential solutions for a quadratic equation, it is crucial to verify them to ensure that they are correct. Verification can be done by substituting the obtained solutions back into the original equation. You check if the substitutions meet the equality stated in the equation.

For our equation, we solved and found \(x = -\frac{10}{3}\) and \(x = \frac{10}{3}\). To verify, substitute \(x = -\frac{10}{3}\) back into \(9x^2 - 100 = 0\):

• Calculate: \(9\left(-\frac{10}{3}\right)^2 - 100 = 0\)

The calculation shows the equality holds true, confirming \(x = -\frac{10}{3}\) is a valid solution.

The same steps show \(x = \frac{10}{3}\) also satisfies the original equation. This verification step ensures that mathematical errors haven’t been made along the way and that the solutions found are completely accurate.