How to Find Factors of Big Numbers in a Quadratic Equation?

Factoring quadratic equations with large numbers can be challenging, but it is definitely possible. Here's a breakdown of how to approach it:

Steps to Factor Quadratic Equations with Large Numbers

Understand the General Form: Remember that a quadratic equation is generally expressed as ax² + bx + c = 0. When factoring, we aim to rewrite it in the form (px + q)(rx + s) = 0.
Focus on the 'c' term (the large number): The key to factoring lies in finding two numbers that multiply to give 'c' and add up* to give 'b'. This is where the challenge arises with larger values of 'c'.
Prime Factorization: Break down the large 'c' value into its prime factors. This will help you systematically identify possible pairs of factors. Example: Let's say c = 72. Its prime factorization is 2 x 2 x 2 x 3 x 3.
Systematically Test Factor Pairs: Use the prime factors to create different pairs of factors for 'c'. For 72, some possible pairs are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9). Consider both positive and negative factors, keeping in mind the sign of 'b'.
Check if the Factors Add Up to 'b': For each pair of factors you create, check if they add up to 'b'. For example, if our quadratic equation was x² + 17x + 72 = 0, we need a pair of factors for 72 that add up to 17. In this case, 8 and 9 work (8 + 9 = 17).
Write the Factored Form: Once you find the correct pair of factors, you can write the quadratic equation in factored form. Using our example: x² + 17x + 72 = (x + 8)(x + 9) = 0.
Solve for x: Finally, set each factor equal to zero and solve for x:
- x + 8 = 0 => x = -8
- x + 9 = 0 => x = -9

Let's factor x² + 38x + 360 = 0

We need factors of 360 that add up to 38.
Prime factorization of 360: 2 x 2 x 2 x 3 x 3 x 5
Start testing pairs. After some trial and error, you'll find that 18 and 20 work: 18 x 20 = 360 and 18 + 20 = 38.
Factored form: (x + 18)(x + 20) = 0
Solutions: x = -18 and x = -20

Start with factors close to the square root of 'c': This often speeds up the process.
Consider the signs of 'b' and 'c': This helps narrow down the possibilities. If 'c' is positive, both factors have the same sign (either both positive or both negative, depending on the sign of 'b'). If 'c' is negative, the factors have opposite signs.
Use a calculator or online tool for prime factorization: This can be a significant time-saver.
Practice: The more you practice, the better you'll become at recognizing factor pairs.