Problem 18 - 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. $$x^{2}-6 x=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^2 - 6x = 0\) are \(x = 0\) and \(x = 6\).

Step 1: Set Equation to Equal Zero

The equation given is \(x^2 - 6x = 0\). It is already in the correct form for factoring.

Step 2: Factor

Factoring the equation results in \(x(x - 6) = 0\). You can get this by taking out the common factor of \(x\) from each term.

Step 3: Solve for \(x\)

A product equals zero if and only if at least one of the factors is zero. Therefore, setting each factor equal to zero yields the solutions \(x = 0\) and \(x - 6 = 0 \rightarrow x = 6\).

Step 4: Substitute the Solutions into the Original Equation

By substituting \(x = 0\) into the original equation, we get \(0^2 - 6*0 = 0\), which is true. Similarly, substituting \(x = 6\) into the original equation, we get \(6^2 - 6*6 = 0\), which is also true. Therefore, the solutions \(x = 0\) and \(x = 6\) are correct.

Key Concepts

Factoring

Factoring is a method of solving quadratic equations by expressing the equation as a product of its factors. A factor is a number or expression that divides another expression without leaving a remainder. In the given quadratic equation \(x^2 - 6x = 0\), our goal is to write it as a product to make solving easier.

To factor this equation, we look for common terms. Both terms in the equation \(x^2\) and \(-6x\) share a factor of \(x\). You can "factor out" this common \(x\), rewriting the equation as \(x(x - 6) = 0\). This process simplifies the quadratic equation into a form where it is easier to find solutions.

Factoring may not always be straightforward, especially for more complex equations, but once factored, it reveals critical points or solutions. Think of factoring as a key step in unlocking the solution of the equation.

Solving Equations

Once an equation is factored, solving it becomes much simpler. You apply a fundamental property of multiplication: if a product of two numbers is zero, then at least one of the numbers must be zero. For the equation \(x(x-6)=0\), this means either \(x=0\) or \(x-6=0\).

- Solving for \(x=0\) is straightforward; \(x\) is simply 0. - For \(x-6 = 0\), add 6 to both sides to isolate \(x\), resulting in \(x=6\).

These solutions represent the values of \(x\) where the original equation equals zero. Checking these values by substituting them back into the original equation ensures accuracy. When substituted back, both \(x=0\) and \(x=6\) satisfy the equation \(x^2 - 6x = 0\), confirming them as correct solutions.

Graphing

Graphing a quadratic equation offers a visual understanding of its solutions. The equation \(x^2 - 6x = 0\) can be graphed as a parabola, a U-shaped curve typical of quadratic equations.

The solutions we found from factoring, \(x = 0\) and \(x = 6\), correspond to the points where the parabola crosses the x-axis, known as the x-intercepts. These intercepts visually confirm the solutions we obtained algebraically, providing a double-check against our calculations.

Using graphing technology, such as a calculator or software, makes this process easier and more accurate. Graphing helps in understanding how changes in the equation affect its solutions and can be particularly helpful when manual graphing is complex. Seeing the parabola and its intersection with the x-axis underlines the relationship between algebraic solutions and their graphical representations.