Solving Quadratic Equations By Factorisation A Step-by-Step Guide

July 10, 2025 by Scholario Team 66 views

Solve Quadratic Equations by Factorising: A Comprehensive Guide

Solving quadratic equations is a fundamental skill in algebra, and one of the most effective methods is factorisation. This article provides a comprehensive guide on how to solve quadratic equations by factorising, with a focus on the equation $2n^2 - 15n + 7 = 0$ . We will break down the steps involved, explain the underlying concepts, and offer tips for mastering this technique. Whether you are a student learning algebra or someone looking to refresh your math skills, this guide will help you understand and apply the factorisation method with confidence.

Understanding Quadratic Equations

Before diving into factorisation, it’s essential to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and x is the variable. The coefficient a cannot be zero; otherwise, the equation becomes linear. The solutions to a quadratic equation are also known as the roots or zeros of the equation. These are the values of x that make the equation true. Quadratic equations can have two, one, or no real solutions, depending on the discriminant ( $b^2 - 4ac$ ).

Key Components of a Quadratic Equation

Quadratic Term: The term $ax^2$ is the quadratic term. It is the term with the variable raised to the power of 2. The coefficient a determines the shape and direction of the parabola when the quadratic equation is graphed.
Linear Term: The term $bx$ is the linear term. It is the term with the variable raised to the power of 1. The coefficient b affects the position and slope of the parabola.
Constant Term: The term c is the constant term. It is the term without any variable. The constant term determines the y-intercept of the parabola.

Importance of Solving Quadratic Equations

Quadratic equations appear in various fields, including physics, engineering, economics, and computer science. They are used to model projectile motion, optimize designs, and analyze financial data. Understanding how to solve quadratic equations is crucial for solving real-world problems. For example, in physics, quadratic equations can describe the trajectory of a ball thrown into the air. In engineering, they can help determine the optimal dimensions of a structure. In finance, they can be used to calculate compound interest.

The Factorisation Method: A Step-by-Step Guide

Factorisation is a method of solving quadratic equations by expressing the quadratic expression as a product of two linear factors. The factorisation method is most effective when the quadratic equation has integer or rational roots. Let's break down the process with our example equation:

$2n^2 - 15n + 7 = 0$

Step 1: Identify the Coefficients

First, identify the coefficients a, b, and c in the equation. In our case:

$a = 2$
$b = -15$
$c = 7$

Step 2: Find Two Numbers

Next, we need to find two numbers that satisfy two conditions:

Their product is equal to $a imes c$ .
Their sum is equal to b.

In our equation:

$a imes c = 2 imes 7 = 14$
$b = -15$

We need to find two numbers that multiply to 14 and add up to -15. These numbers are -14 and -1. You can systematically list the factor pairs of 14 (1 and 14, 2 and 7) and consider their negative counterparts to find the correct pair.

Step 3: Rewrite the Middle Term

Rewrite the middle term (bx) using the two numbers we found. In our case, we replace -15n with -14n - n:

$2n^2 - 14n - n + 7 = 0$

Step 4: Factor by Grouping

Now, we factor by grouping. Group the first two terms and the last two terms:

$(2n^2 - 14n) + (-n + 7) = 0$

Factor out the greatest common factor (GCF) from each group:

$2n(n - 7) - 1(n - 7) = 0$

Notice that both terms now have a common factor of $(n - 7)$ . Factor this out:

$(2n - 1)(n - 7) = 0$

Step 5: Set Each Factor Equal to Zero

To find the solutions, set each factor equal to zero:

$2n - 1 = 0$
$n - 7 = 0$

Step 6: Solve for n

Solve each equation for n:

$2n - 1 = 0 ightarrow 2n = 1 ightarrow n = rac{1}{2}$
$n - 7 = 0 ightarrow n = 7$

Thus, the solutions to the equation $2n^2 - 15n + 7 = 0$ are $n = rac{1}{2}$ and $n = 7$ .

Tips for Mastering Factorisation

Factorisation can be challenging at first, but with practice, you can master this technique. Here are some tips to help you:

Practice Regularly: The more you practice, the better you will become at identifying patterns and factoring quadratic expressions. Work through a variety of examples to build your skills.
Understand the Basics: Make sure you have a solid understanding of the basic algebraic operations and factoring techniques. Review factoring common factors, difference of squares, and perfect square trinomials.
Use Systematic Methods: Develop a systematic approach to finding the two numbers needed for factorisation. Listing factor pairs and checking their sums can help you find the correct pair efficiently.
Check Your Work: After factorising, multiply the factors back together to ensure you get the original quadratic expression. This helps you catch any errors.
Recognize Special Cases: Be aware of special cases like difference of squares ( $a^2 - b^2 = (a + b)(a - b)$ ) and perfect square trinomials ( $a^2 + 2ab + b^2 = (a + b)^2$ ). Recognizing these patterns can simplify factorisation.

Alternative Methods for Solving Quadratic Equations

While factorisation is a powerful method, it is not always the most efficient or practical. There are other methods for solving quadratic equations, including:

Quadratic Formula: The quadratic formula can be used to solve any quadratic equation, regardless of whether it can be easily factorised. The formula is:

$x = rac{-b eq ext{√}(b^2 - 4ac)}{2a}$

The quadratic formula is especially useful when the roots are irrational or complex.
Completing the Square: Completing the square is a method that involves rewriting the quadratic equation in the form $(x - h)^2 = k$ . This method can be used to derive the quadratic formula and is also useful for understanding the vertex form of a quadratic equation.

When to Use Factorisation

Factorisation is most effective when:

The quadratic equation has integer or rational roots.
The coefficients are relatively small and easy to work with.
You can quickly identify the factors.

If the roots are irrational or complex, or if the coefficients are large or complicated, the quadratic formula or completing the square may be more efficient.

Common Mistakes to Avoid

When solving quadratic equations by factorising, it’s important to avoid common mistakes. Here are some pitfalls to watch out for:

Incorrectly Identifying Coefficients: Make sure you correctly identify the coefficients a, b, and c. A mistake in this step can lead to incorrect factorisation.
Errors in Finding the Two Numbers: Ensure that the two numbers you find multiply to $a imes c$ and add up to b. Double-check your calculations to avoid errors.
Incorrect Factoring by Grouping: When factoring by grouping, make sure you factor out the greatest common factor correctly and that the remaining terms in the parentheses are the same.
Forgetting to Set Factors to Zero: Remember to set each factor equal to zero and solve for the variable. This is a crucial step in finding the solutions.
Not Checking Your Answers: Always check your solutions by substituting them back into the original equation. This will help you catch any mistakes and ensure your answers are correct.

Real-World Applications

Understanding how to solve quadratic equations by factorising has numerous real-world applications. Here are a few examples:

Physics: Quadratic equations are used to model projectile motion, such as the trajectory of a ball thrown into the air. By solving quadratic equations, we can determine the maximum height, range, and time of flight of the projectile.
Engineering: Engineers use quadratic equations to design structures, calculate stress and strain, and optimize designs. For example, quadratic equations can be used to determine the dimensions of a bridge or the shape of an arch.
Finance: Quadratic equations are used in finance to calculate compound interest, determine the break-even point for a business, and model investment returns. Understanding quadratic equations can help make informed financial decisions.
Computer Science: Quadratic equations are used in computer graphics, game development, and algorithm design. For example, they can be used to calculate the intersection of lines and curves or to optimize search algorithms.

Conclusion

Solving quadratic equations by factorisation is a fundamental skill in algebra with wide-ranging applications. By understanding the steps involved, practicing regularly, and avoiding common mistakes, you can master this technique. In this guide, we have covered the step-by-step process of factorisation, from identifying coefficients to setting factors to zero. We have also provided tips for mastering factorisation, discussed alternative methods for solving quadratic equations, and highlighted real-world applications. Remember, consistent practice and a solid understanding of the underlying concepts are key to success in solving quadratic equations. Whether you're tackling homework assignments or applying algebra in real-life scenarios, the ability to factorise quadratic equations will prove to be an invaluable asset.

By following this comprehensive guide, you should now be well-equipped to solve quadratic equations by factorising, including the example equation $2n^2 - 15n + 7 = 0$ . Keep practicing, and you'll become more confident and proficient in this essential algebraic skill. Remember, factorising is not just a mathematical exercise; it's a powerful tool that can help you solve problems in various fields and make informed decisions in real-world situations.