Problem 99 - Elementary Algebra

Question

For exercises 97-100, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(x^{2}+12 x=0\) Incorrect Answer: \(x^{2}+12 x=0\) $$ \begin{aligned} \frac{x^{2}}{x}+\frac{12 x}{x} &=\frac{0}{x} \\ x+12 &=0 \\ \frac{-12}{x} &-12 \\ \hline x &=-12 \end{aligned} $$ Solution: \(x=-12\)

Step-by-Step Solution

Answer

The correct solutions are x = 0 and x = -12.

Step 1: Identify the Mistake

The mistake in the problem occurs in the step where both terms on the left side of the equation are divided by x. This division is incorrect because it skips a valid solution where x equals 0.

Step 1: Factor the Equation

Rewrite the equation in its factored form. The original equation is given by:\[ x^2 + 12x = 0 \] Factor out the common term x: \[ x(x + 12) = 0 \]

Step 2: Solve for x

Set each factor equal to zero and solve for x: \[ x = 0 \] \[ x + 12 = 0 \] \[ x = -12 \]

Step 3: Verify Solutions

The solutions to the equation are: \[ x = 0 \] and \[ x = -12 \] Verify each solution by substituting back into the original equation.

Key Concepts

quadratic equation

A quadratic equation is a second-degree polynomial equation in the form of:
\[ ax^2 + bx + c = 0 \]
Where:

• '\(a\)' is the coefficient of the squared term,
• '\(b\)' is the coefficient of the linear term,
• '\(c\)' is the constant term.

Quadratic equations often have two solutions or roots. These roots can be real or complex numbers.
Solving a quadratic equation means finding the values of \(x\) that make the equation true.

In our exercise, the quadratic equation was: \(x^2 + 12x = 0\)

factoring

Factoring a quadratic equation involves expressing it as a product of its linear factors, if possible.
This method is commonly used because it simplifies solving the equation.
Let's consider the exercise we were working on:
\[ x^2 + 12x = 0 \]
The first step is to look for common factors in each term. In this case, 'x' is a common factor:
\[ x(x + 12) = 0 \]
By factoring, we transform the quadratic equation into a product of simpler expressions. This makes it easier to find the solutions.

solutions of equations

Once we have the factored form of the quadratic equation, we use the Zero Product Property, which states:
If \(a \times b = 0\), then either \(a = 0\) or \(b = 0\), or both.

Applying this property to the factored equation:
\[ x(x + 12) = 0 \]
we set each factor to zero and solve for 'x':

• '\(x = 0\)',
• '\(x + 12 = 0\)' which gives us '\(x = -12\)'.

Therefore, the solutions to the quadratic equation \(x^2 + 12x = 0\) are \(x = 0\) and \(x = -12\). By factoring, we have found both solutions.

verifying solutions

Verifying solutions means substituting the found solutions back into the original equation to check if they satisfy the equation.
This step ensures that our solutions are correct.
Let's verify the solutions we found: \(x = 0\) and \(x = -12\):

For \(x = 0\):
\[ (0)^2 + 12(0) = 0 \]
which simplifies to \(0 = 0\) (True).
For \(x = -12\):
\[ (-12)^2 + 12(-12) = 0 \]
which simplifies to \(144 - 144 = 0\) (True).
Both solutions satisfy the original equation, confirming they are correct. Always verify solutions to ensure the problem is solved correctly.