Quadratic equations are one of the most important topics in algebra. They appear in physics, engineering, finance, and even in everyday problem-solving. While there are different ways to solve quadratic equations, the factorization method is often the simplest and most straightforward.
In this article, weβll explain what quadratic equations are, walk through the factorization method step by step, and provide lots of examples so you can master this technique. After leading you to formula method, we will be discussing the factorization method.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in the form:
ax2 + bx + c = 0
where:
- a, b, and c are real numbers,
- a β 0,
- x is the unknown variable.
For example:
x2 + 5x + 6 = 0
This is a quadratic equation because the highest power of x is 2.
What is the Factorization Method?
The factorization method means rewriting a quadratic equation as the product of two linear expressions (binomials).
If:
ax2 + bx + c = 0
can be expressed as:
(x + p) (x + q) = 0
then, by the zero product property,
x + p = 0 or x + q = 0
So the solutions are:
X = βp, x = βq
Also Read: How to Solve Quadratic Equations Using the Formula Method
Step-by-Step Guide to Solving by Factorization
- Write the quadratic equation in standard form:
ax2 + bx + c = 0
- Multiply π and π (the first and last coefficients).
- Find two numbers that multiply to give πΓπ and add to give π.
- Split the middle term using these two numbers.
- Group terms in pairs and factorize.
- Solve each factor by setting it equal to zero.
Examples of Solving Quadratics by Factorization
Example 1: Simple Quadratic
x2 + 5x + 6 = 0
Step 1:
A = 1, b = 5, c = 6
Step 2: Multiply
A Γ c = 1 Γ 6 = 6
Step 3: Find two numbers whose product = 6 and sum = 5 β 2 and 3
Step 4: Rewrite the equation:
x2 + 2x + 3x + 6 = 0
Step 5: Group terms:
(x2 + 2x) + (3x + 6) = 0
Factorize:
x(x + 2) + 3(x + 2) = 0
(x + 2) (x + 3) = 0
Step 6: Solve each factor:
X + 2 = 0 β x = β2
X + 3 = 0 β x = β3
Final Answer: X = β2, β3
Example 2: With Coefficient of π₯2
2x2 + 7x + 3 = 0
Step 1:
A = 2, b = 7, c = 3
Step 2: Multiply π Γ c = 2 Γ 3 = 6
Step 3: Find two numbers whose product = 6 and sum = 7 β 6 and 1
Step 4: Split the middle term:
2x2 + 6x + x + 3 = 0
Step 5: Group:
(2x2 + 6x) + (x + 3) = 0
2x(x + 3) + 1(x + 3) = 0
(2x + 1) (x + 3) = 0
Step 6: Solve:
2x + 1 = 0 β x = β1/2
X + 3 = 0 β x = β3
Final Answer: x = β1/2, β3
Example 3: Negative Coefficients
x2 β 8x + 12 = 0
Step 1: A = 1, b = β8, c = 12
Step 2: Multiply a Γ c = 12
Step 3: Numbers whose product = 12 and sum = -8 β -6 and -2
Step 4: Split:
x2 β 6x β 2x + 12 = 0
Step 5: Group:
(x2 β 6x) β (2x β 12) = 0
x(x β 6) β 2(x β 6) = 0
(x β 6) (x β 2) = 0
Step 6: Solve:
X = 6, x = 2
Final Answer:
x = 2, 6
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Practice Problems
Try solving these on your own using the factorization method:
x2 + 9x + 14 = 0
3x2 + 11x + 6 = 0
x2 β 5x + 6 = 0
2x2 β x β15 = 0
(Hint: Follow the steps carefully: multiply πΓπ, split the middle term, and factorize).
Conclusion
The factorization method is one of the easiest ways to solve quadratic equations when they can be expressed as the product of two binomials. By following the stepsβmultiplying, splitting the middle term, grouping, and factorizingβyou can solve quadratic equations quickly and accurately without any stress. Mastering this method not only builds a strong foundation in algebra but also helps in higher-level math and real-world problem solving.