Problem 17 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ (x+7)(x-7)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -7\) and \(x = 7\).

Step 1: Apply the Zero Product Property

The Zero Product Property states that if the product of two expressions is zero, then at least one of the expressions must be zero. For the equation \((x+7)(x-7)=0\), this means either \(x+7=0\) or \(x-7=0\).

Step 2: Solve the First Equation

First, solve the equation \(x+7=0\). Subtract 7 from both sides to isolate \(x\). \(x = -7\).

Step 3: Solve the Second Equation

Next, solve the equation \(x-7=0\). Add 7 to both sides to isolate \(x\). \(x = 7\).

Step 4: Write the Solution Set

The solutions to the equation \((x+7)(x-7)=0\) are \(x = -7\) and \(x = 7\). Write the solution set as \(\{-7, 7\}\).

Key Concepts

Quadratic Equations

Quadratic equations form the foundation of a wide array of mathematical concepts. A quadratic equation is any equation that can be rearranged in the standard form:

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants and \( x \) represents an unknown variable.
The most distinctive feature of a quadratic equation is the \( x^2 \) term, which makes the equation a polynomial of degree 2.
This means the graph of a quadratic equation is always a parabola. Quadratic equations can have zero, one, or two real solutions, depending on the value of the discriminant \( b^2 - 4ac \).

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is one real solution.
• If it is negative, there are no real solutions, but two complex solutions instead.

Understanding these solutions and how to find them is critical in both pure and applied mathematics, making mastering quadratic equations an essential skill for any student.

Factoring

Factoring is an essential technique used in solving quadratic equations and simplifying algebraic expressions.
When you "factor" a quadratic expression, you are breaking it down into simpler expressions that, when multiplied together, yield the original expression.
This process is akin to translating a complex sentence into simpler statements.For many simple quadratics, this involves finding two binomials that multiply to give the original quadratic.
For example, in the equation \((x + 7)(x - 7) = 0\), the expression is already factored.
Each binomial represents a factor of the quadratic, offering a direct path to finding solutions using the Zero Product Property. The key to effective factoring is practice, as recognizing patterns and knowing various factoring techniques can significantly speed up solving problems.
The most common methods include:

• Factoring by grouping
• The difference of squares
• Completing the square

Mastery of these techniques helps unlock the potential to solve more complex algebraic problems with ease.

Algebraic Solutions

Algebraic solutions involve finding values for variables that satisfy given algebraic equations.
In the case of quadratic equations, the algebraic approach primarily involves manipulating the equation to isolate the variable.
This manipulation often requires a combination of simpler operations, such as addition, subtraction, and sometimes division or multiplication.When an equation like \((x+7)(x-7)=0\) is given, we can employ the Zero Product Property as a powerful tool.
This property tells us that if a product of two expressions is zero, then one or both of the expressions must also be zero.

• For \(x+7=0\), solve for \(x\) by subtracting 7, leading to \(x=-7\).
• For \(x-7=0\), solve for \(x\) by adding 7, resulting in \(x=7\).

The combination of these solutions gives us the solution set \(\{-7, 7\}\).
Understanding how to solve these equations algebraically helps students build stronger problem-solving skills and prepares them for advanced math courses.