Quadratic Equations by Factorization: 10 Solved Examples

Remember that feeling when you're staring at a math problem, and it just won't click? Especially in Class 10, when every mark feels super important for your board exams. Quadratic equations, yaar, can sometimes feel like that tricky puzzle, right?

But what if I told you there's a straightforward, super effective method that can help you crack almost any quadratic equation? That's right, we're talking about the factorization method, or as many of you know it, 'splitting the middle term'. It's a foundational skill that will serve you well, not just in Class 10 but also in higher studies and competitive exams.

Did you know that 40% of CBSE Class 10 students score below 60% in math? A big reason is often not mastering fundamental topics like quadratic equations. So, let's change that statistic for you, starting today!

Before we dive into solving, let's quickly recap what a quadratic equation is. It's any equation that can be written in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers, and $a
eq 0$. The 'degree' of the equation is 2, meaning the highest power of the variable (usually$x$) is 2.

For my CBSE friends, this is a crucial topic from Chapter 4 of your NCERT textbook, often carrying 4-6 marks in the board exams. Mastering it will help you ace those questions, especially the ones from RD Sharma or RS Aggarwal that you practice daily.

For my ICSE buddies, you know Selina Concise and S.Chand really push conceptual understanding. While ICSE Math often has a higher difficulty level than CBSE, it also offers better conceptual depth, and factorization is a core skill you'll build on. This method is fundamental for your internal assessments too!

The factorization method relies on a simple idea: if the product of two numbers is zero, then at least one of them must be zero. This is called the Zero Product Property. Our goal is to transform the quadratic equation into a product of two linear factors.

Here’s how we do it, step-by-step, using the 'splitting the middle term' technique:

1. Standard Form: Make sure your quadratic equation is in the standard form: $ax^2 + bx + c = 0$.
2. Find the Product: Multiply the coefficient of $x^2$ ($a$) and the constant term ($c$). Let this product be $P = a \times c$.
3. Split the Middle Term: Find two numbers, let's call them $p$ and $q$, such that their sum is equal to the coefficient of $x$ ($b$), i.e., $p + q = b$, AND their product is equal to $P$ (from step 2), i.e., $p \times q = ac$.
4. Rewrite the Equation: Rewrite the middle term $bx$ as $px + qx$. So, the equation becomes $ax^2 + px + qx + c = 0$.
5. Factor by Grouping: Group the first two terms and the last two terms, then factor out the common monomial from each pair.
6. Apply Zero Product Property: You'll end up with an equation like $(dx + e)(fx + g) = 0$. Now, set each factor equal to zero and solve for $x$. These values of $x$ are the roots (or solutions) of your quadratic equation. Simple, accha?

Now that you know the steps, let's get our hands dirty with some worked examples. We'll go through each one carefully, so you can see the method in action. Pay close attention to the signs, they are super important!

Example 1: A Straightforward Start

Solve for $x$: $x^2 + 5x + 6 = 0$

Solution:

1. Standard Form: The equation is already in standard form ($a=1, b=5, c=6$).
2. **Find the Product ($ac$):** $a \times c = 1 \times 6 = 6$.
3. **Split the Middle Term ($b=5$):** We need two numbers whose sum is 5 and product is 6. These numbers are 2 and 3 ($2+3=5$ and $2 \times 3=6$).
4. Rewrite the Equation: Substitute $5x$ with $2x + 3x$:

Example 2: Dealing with Negative Signs

Solve for $x$: $x^2 - 7x + 12 = 0$

Solution:

1. Standard Form: The equation is in standard form ($a=1, b=-7, c=12$).
2. **Find the Product ($ac$):** $a \times c = 1 \times 12 = 12$.
3. **Split the Middle Term ($b=-7$):** We need two numbers whose sum is -7 and product is 12. Since the product is positive and the sum is negative, both numbers must be negative. These numbers are -3 and -4 ($-3 + (-4) = -7$ and $(-3) \times (-4) = 12$).
4. Rewrite the Equation: Substitute $-7x$ with $-3x - 4x$:

Example 3: Coefficients Greater Than One

Solve for $x$: $2x^2 + 7x + 3 = 0$

Solution:

1. Standard Form: Already in standard form ($a=2, b=7, c=3$).
2. **Find the Product ($ac$):** $a \times c = 2 \times 3 = 6$.
3. **Split the Middle Term ($b=7$):** We need two numbers whose sum is 7 and product is 6. These numbers are 1 and 6 ($1+6=7$ and $1 \times 6 = 6$).
4. Rewrite the Equation: Substitute $7x$ with $x + 6x$:

Example 4: A Word Problem Scenario

The product of two consecutive positive integers is 306. Find the integers.

Solution:

1. Formulate the Equation: Let the first positive integer be $x$. Since the integers are consecutive, the next integer will be $x+1$. Their product is 306.

You might be thinking, "Is this just for exams?" Absolutely not! Quadratic equations are everywhere around us, from sports to engineering.

Ever wondered about the path a cricket ball takes when hit for a six? That's a parabola, described by a quadratic equation! Engineers use them to design bridges, architects use them for arches, and even economists use them to model profit and loss functions. So, understanding quadratics isn't just about scoring marks; it's about understanding the world!

Let's quickly recap what we've learned:

* Quadratic equations are of the form $ax^2 + bx + c = 0$.
* Factorization (splitting the middle term) is a powerful method to find the roots.
* The core idea is to find two numbers ($p, q$) such that $p+q=b$ and $pq=ac$.
* Always remember the Zero Product Property: if $(X)(Y)=0$, then $X=0$ or $Y=0$.
* Practice is essential for mastering this method and improving speed and accuracy.
* Quadratic equations have many real-world applications in science, engineering, and economics.