Solving Quadratic Equations Using The Zero Product Property

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The question presents a quadratic equation, $z^2 - 15z + 56 = 0$, and asks us to solve it using the zero product property. This property is a fundamental concept in algebra and provides an efficient method for finding the roots (or solutions) of quadratic equations that can be factored. In this article, we will delve into the step-by-step process of solving the given equation using the zero product property, while also exploring the underlying principles and broader applications of this important algebraic tool.

Understanding the Zero Product Property

The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. Mathematically, this can be expressed as follows:

If $a * b = 0$, then $a = 0$ or $b = 0$ (or both).

This seemingly simple property is the cornerstone of solving many algebraic equations, especially quadratic equations. Quadratic equations are polynomial equations of the second degree, generally expressed in the form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and x is the variable. The zero product property allows us to transform a quadratic equation in this standard form into a simpler form that is easier to solve.

Step-by-Step Solution

Now, let's apply the zero product property to solve the given equation:

$z^2 - 15z + 56 = 0$

1. Factor the Quadratic Expression

The first step is to factor the quadratic expression $z^2 - 15z + 56$. Factoring involves expressing the quadratic as a product of two binomials. We need to find two numbers that:

• Multiply to give the constant term (56).
• Add up to give the coefficient of the linear term (-15).

By considering the factors of 56, we can identify the numbers 8 and 7. Since the coefficient of the linear term is -15, both numbers should be negative. Thus, -8 and -7 satisfy the conditions:

• (-8) * (-7) = 56
• (-8) + (-7) = -15

Therefore, we can factor the quadratic expression as follows:

$z^2 - 15z + 56 = (z - 8)(z - 7)$

2. Apply the Zero Product Property

Now that we have factored the quadratic equation, we can rewrite the original equation as:

$(z - 8)(z - 7) = 0$

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for z:

$z - 8 = 0$ or $z - 7 = 0$

3. Solve for z

Solving each equation separately:

• For $z - 8 = 0$, add 8 to both sides to get $z = 8$.
• For $z - 7 = 0$, add 7 to both sides to get $z = 7$.

Thus, the solutions to the quadratic equation are $z = 8$ and $z = 7$.

The Correct Answer

Comparing our solutions with the given options:

A) $z = -4, 7$ B) $z = 8, 7$ C) $z = 8, -4$ D) $z = -5, -2$

We can see that option B, $z = 8, 7$, matches our solutions. Therefore, the correct answer is B.

Importance of the Zero Product Property

The zero product property is a cornerstone in solving quadratic equations and other polynomial equations. It simplifies the process of finding roots by breaking down a complex equation into simpler, linear equations. This property is not only applicable to quadratic equations but also extends to higher-degree polynomials, making it a versatile tool in algebra.

Applications Beyond Quadratic Equations

The zero product property is used extensively in various areas of mathematics, including:

• Solving polynomial equations of higher degrees.
• Finding the x-intercepts (or zeros) of polynomial functions.
• Solving systems of equations.
• Simplifying algebraic expressions.

Its ability to transform complex problems into manageable parts makes it an indispensable tool for students and professionals alike.

Alternative Methods for Solving Quadratic Equations

While the zero product property is effective for solving factorable quadratic equations, other methods can be used, especially when the equation is not easily factorable. These methods include:

1. Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of whether it can be factored. The formula is given by:

$z = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

where a, b, and c are the coefficients of the quadratic equation in the standard form $az^2 + bz + c = 0$. This formula provides the roots of the equation, and it is particularly useful when the equation is difficult to factor or has irrational roots.

2. Completing the Square

Completing the square is another method for solving quadratic equations. It involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly useful for understanding the structure of quadratic equations and for deriving the quadratic formula.

3. Graphing

Graphing the quadratic equation can also provide approximate solutions. The roots of the equation correspond to the x-intercepts of the graph. While this method may not give exact solutions, it offers a visual representation of the roots and can be helpful in understanding the behavior of quadratic functions.

Common Mistakes to Avoid

When using the zero product property, it is essential to avoid common mistakes that can lead to incorrect solutions. Some of these mistakes include:

1. Incorrect Factoring

The most common mistake is incorrect factoring of the quadratic expression. It is crucial to ensure that the factors multiply to give the original quadratic expression. Always double-check your factors by expanding them to verify that they match the original expression.

2. Not Setting Each Factor to Zero

Another mistake is forgetting to set each factor equal to zero. The zero product property states that if the product of factors is zero, then each factor must be considered. Failing to set each factor to zero will result in missing solutions.

3. Dividing by a Variable Expression

Avoid dividing both sides of the equation by a variable expression. This can lead to the loss of solutions. For example, if you have the equation $z(z - 5) = 0$, dividing both sides by z would eliminate the solution $z = 0$.

4. Misapplying the Quadratic Formula

When using the quadratic formula, ensure that you correctly identify the coefficients a, b, and c, and substitute them accurately into the formula. Mistakes in substitution can lead to incorrect solutions.

Conclusion

In summary, solving the quadratic equation $z^2 - 15z + 56 = 0$ using the zero product property involves factoring the quadratic expression into $(z - 8)(z - 7)$, setting each factor equal to zero, and solving for z. This yields the solutions $z = 8$ and $z = 7$, making option B the correct answer. The zero product property is a powerful tool in algebra, applicable to a wide range of equations, and understanding its principles and applications is crucial for mastering algebraic problem-solving. By avoiding common mistakes and practicing regularly, you can confidently solve quadratic equations and other polynomial equations using this fundamental property.

This article has provided a detailed explanation of how to solve a quadratic equation using the zero product property, along with insights into its broader applications and alternative methods for solving quadratic equations. By understanding these concepts, you can enhance your problem-solving skills and excel in algebra.