91Ó°ÊÓ

When dealing with quadratic equations, a common method of solving them is factoring. Quadratic equations are typically in the form of \( ax^2 + bx + c = 0 \). Our task is to express it as a product of two binomials. Once factored, solving the equation becomes straightforward.

In the equation \( x^2 - 6x - 7 = 0 \), we specifically look for two numbers that multiply to \(-7\) and add up to \(-6\). These are the coefficients from which we will derive our binomials. For this equation:

• - Multiplying to \(-7\) comes from the constant term \(c\).
• - Adding to \(-6\) comes from the middle term \(b\).

Once identified, these numbers are \(-7\) and \(1\), leading to the factorization: \( (x - 7)(x + 1) = 0 \).

This step is crucial, as it creates the pathway to our solutions.

The Zero Product Property is a fundamental principle used when solving factored quadratic equations. It states that if the product of two numbers is zero, at least one of the multiplicands must be zero. This property allows us to solve for \(x\) once we have factored the quadratic equation.

For \( (x - 7)(x + 1) = 0 \), we apply this property by setting each factor equal to zero:

• \(x - 7 = 0\)
• \(x + 1 = 0\)

Solving these individual equations gives us the values of \(x\) that satisfy the original quadratic equation:
, ,
By employing this property, we break down a potentially complex quadratic problem into two much simpler linear equations.

Solving quadratic equations is a skill that opens the door to understanding a wide range of mathematical concepts. The ultimate goal of solving a quadratic equation is to find the values of \(x\) that satisfy the equation.

For the equation \( x^2 - 6x - 7 = 0 \), we found the solutions through factoring and applying the Zero Product Property:

• The initial step involved recognizing the equation as quadratic.
• Next, we factored it into \( (x - 7)(x + 1) = 0 \).
• We then applied the Zero Product Property to find \(x = 7\) and \(x = -1\).

To finalize, it's important to verify these solutions by substituting them back into the original equation. If both actual solutions satisfy \(x^2 - 6x - 7 = 0\), then the solutions are correct. Indeed, substituting confirms that both solutions work, ensuring the accuracy of our work.

Understanding these processes builds a strong foundation for further mathematical exploration.