Factoring a Quadratic Expression by Removing a Common Factor

If you're working on factoring a quadratic expression, it's important to look for the greatest common factor (GCF) first. Once you remove the GCF, you can attempt to factor the remaining expression. Let's explore this process using the given quadratic expression: 3x2 - 21x - 36.

Step 1: Identify the GCF

First, we need to determine the GCF of the coefficients in the expression. Here, the coefficients are 3, -21, and -36. The GCF of these numbers is 3. We can factor out the GCF from the expression:

3x2 - 21x - 36 3(x2 - 7x - 12)

Now, we have a simpler expression inside the parentheses: x2 - 7x - 12.

Step 2: Factor the Quadratic Expression

Next, we need to factor the quadratic expression x2 - 7x - 12. We'll use the method of finding two numbers that multiply to -12 and add up to -7. These two numbers are -12 and 5 because (-12) * 5 -12 and -12 5 -7.

x2 - 7x - 12 (x - 12)(x 5)

So, we can rewrite the original expression as:

3x2 - 21x - 36 3(x - 12)(x 5)

Alternative: Solving with the Quadratic Formula

If factoring is not straightforward, you can solve the quadratic equation by using the quadratic formula. For the equation 3x2 - 21x - 36 0, the quadratic formula is:

x [-b ± √(b2 - 4ac)] / 2a

Here, a 3, b -21, and c -36. Plugging these values into the formula, we get:

x [21 ± √((-21)2 - 4 * 3 * -36)] / 2 * 3

x [21 ± √(441 432)] / 6

x [21 ± √873] / 6

This will give you the roots of the quadratic equation, which can be solved using a calculator.

Conclusion

The process of factoring a quadratic expression by removing a common factor is a valuable skill in algebra. Always start by looking for the GCF before attempting more complex factoring techniques. If the expression resists factoring, the quadratic formula can be used to find the roots.

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