Mastering Factoring Quadratic Expressions A Step By Step Guide

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Hey guys! Factoring can seem tricky, but it's a super important skill in algebra. Let's break down some quadratic expressions and get comfortable with factoring. We'll work through several examples step-by-step. Trust me, once you get the hang of it, it's almost like solving a puzzle!

Why is Factoring Important?

Before we dive into the problems, let's quickly talk about why factoring matters. Factoring is essentially the reverse process of expanding. When we expand, we multiply out expressions like (x+2)(x+3)(x + 2)(x + 3) to get x2+5x+6x^2 + 5x + 6. Factoring takes us the other way – we start with x2+5x+6x^2 + 5x + 6 and want to find the original factors, (x+2)(x + 2) and (x+3)(x + 3).

Factoring is crucial for:

  • Solving quadratic equations: Many quadratic equations can be easily solved once they are factored.
  • Simplifying algebraic expressions: Factoring can help us cancel out common factors in fractions.
  • Graphing functions: The factored form of a quadratic equation tells us the x-intercepts of the parabola.

So, with that in mind, let's get started!

Factoring Quadratic Expressions: Examples

We'll be focusing on factoring quadratic expressions of the form ax2+bx+cax^2 + bx + c. There are different techniques we can use, but a common one involves finding two numbers that multiply to give acac and add up to bb. Let's see how this works in practice.

1) Factoring $2x^2 + 3x - 2$

Okay, let's tackle the first one: $2x^2 + 3x - 2$.

Step 1: Identify a, b, and c

In this case, a=2a = 2, b=3b = 3, and c=βˆ’2c = -2.

Step 2: Calculate ac

Multiply aa and cc: 2βˆ—βˆ’2=βˆ’42 * -2 = -4.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to -4 and add up to 3. Those numbers are 4 and -1.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing 3x3x with 4xβˆ’x4x - x: $2x^2 + 4x - x - 2$

Step 5: Factor by grouping

Group the first two terms and the last two terms: $(2x^2 + 4x) + (-x - 2)$

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

2x(x+2)βˆ’1(x+2)2x(x + 2) - 1(x + 2)

Notice that we now have a common factor of (x+2)(x + 2).

Step 6: Factor out the common factor

Factor out (x+2)(x + 2): $(2x - 1)(x + 2)$

So, the factored form of $2x^2 + 3x - 2$ is $(2x - 1)(x + 2)$.

2) Factoring $3x^2 - 5x - 2$

Let's try another one: $3x^2 - 5x - 2$

Step 1: Identify a, b, and c

Here, a=3a = 3, b=βˆ’5b = -5, and c=βˆ’2c = -2.

Step 2: Calculate ac

Multiply aa and cc: 3βˆ—βˆ’2=βˆ’63 * -2 = -6.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to -6 and add up to -5. Those numbers are -6 and 1.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing βˆ’5x-5x with βˆ’6x+x-6x + x: $3x^2 - 6x + x - 2$

Step 5: Factor by grouping

Group the terms: $(3x^2 - 6x) + (x - 2)$

Factor out the GCF from each group:

3x(xβˆ’2)+1(xβˆ’2)3x(x - 2) + 1(x - 2)

Step 6: Factor out the common factor

Factor out (xβˆ’2)(x - 2): $(3x + 1)(x - 2)$

Therefore, the factored form of $3x^2 - 5x - 2$ is $(3x + 1)(x - 2)$.

3) Factoring $6x^2 + 7x + 2$

Time for another example: $6x^2 + 7x + 2$

Step 1: Identify a, b, and c

In this case, a=6a = 6, b=7b = 7, and c=2c = 2.

Step 2: Calculate ac

Multiply aa and cc: 6βˆ—2=126 * 2 = 12.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to 12 and add up to 7. Those numbers are 3 and 4.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing 7x7x with 3x+4x3x + 4x: $6x^2 + 3x + 4x + 2$

Step 5: Factor by grouping

Group the terms: $(6x^2 + 3x) + (4x + 2)$

Factor out the GCF from each group:

3x(2x+1)+2(2x+1)3x(2x + 1) + 2(2x + 1)

Step 6: Factor out the common factor

Factor out (2x+1)(2x + 1): $(3x + 2)(2x + 1)$

So, the factored form of $6x^2 + 7x + 2$ is $(3x + 2)(2x + 1)$.

4) Factoring $5x^2 + 13x - 6$

Let's keep the ball rolling: $5x^2 + 13x - 6$

Step 1: Identify a, b, and c

Here, a=5a = 5, b=13b = 13, and c=βˆ’6c = -6.

Step 2: Calculate ac

Multiply aa and cc: 5βˆ—βˆ’6=βˆ’305 * -6 = -30.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to -30 and add up to 13. Those numbers are 15 and -2.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing 13x13x with 15xβˆ’2x15x - 2x: $5x^2 + 15x - 2x - 6$

Step 5: Factor by grouping

Group the terms: $(5x^2 + 15x) + (-2x - 6)$

Factor out the GCF from each group:

5x(x+3)βˆ’2(x+3)5x(x + 3) - 2(x + 3)

Step 6: Factor out the common factor

Factor out (x+3)(x + 3): $(5x - 2)(x + 3)$

Therefore, the factored form of $5x^2 + 13x - 6$ is $(5x - 2)(x + 3)$.

5) Factoring $6x^2 - 5x - 6$

On to the next one: $6x^2 - 5x - 6$

Step 1: Identify a, b, and c

In this case, a=6a = 6, b=βˆ’5b = -5, and c=βˆ’6c = -6.

Step 2: Calculate ac

Multiply aa and cc: 6βˆ—βˆ’6=βˆ’366 * -6 = -36.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to -36 and add up to -5. Those numbers are -9 and 4.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing βˆ’5x-5x with βˆ’9x+4x-9x + 4x: $6x^2 - 9x + 4x - 6$

Step 5: Factor by grouping

Group the terms: $(6x^2 - 9x) + (4x - 6)$

Factor out the GCF from each group:

3x(2xβˆ’3)+2(2xβˆ’3)3x(2x - 3) + 2(2x - 3)

Step 6: Factor out the common factor

Factor out (2xβˆ’3)(2x - 3): $(3x + 2)(2x - 3)$

So, the factored form of $6x^2 - 5x - 6$ is $(3x + 2)(2x - 3)$.

6) Factoring $12x^2 - x - 6$

Alright, let's keep going! Next up is $12x^2 - x - 6$

Step 1: Identify a, b, and c

Here, a=12a = 12, b=βˆ’1b = -1, and c=βˆ’6c = -6.

Step 2: Calculate ac

Multiply aa and cc: 12βˆ—βˆ’6=βˆ’7212 * -6 = -72.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to -72 and add up to -1. Those numbers are -9 and 8.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing βˆ’x-x with βˆ’9x+8x-9x + 8x: $12x^2 - 9x + 8x - 6$

Step 5: Factor by grouping

Group the terms: $(12x^2 - 9x) + (8x - 6)$

Factor out the GCF from each group:

3x(4xβˆ’3)+2(4xβˆ’3)3x(4x - 3) + 2(4x - 3)

Step 6: Factor out the common factor

Factor out (4xβˆ’3)(4x - 3): $(3x + 2)(4x - 3)$

Therefore, the factored form of $12x^2 - x - 6$ is $(3x + 2)(4x - 3)$.

7) Factoring $4a^2 + 15a + 9$

Let's switch things up a bit and use 'a' instead of 'x': $4a^2 + 15a + 9$

Step 1: Identify a, b, and c

In this case, a=4a = 4, b=15b = 15, and c=9c = 9.

Step 2: Calculate ac

Multiply aa and cc: 4βˆ—9=364 * 9 = 36.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to 36 and add up to 15. Those numbers are 12 and 3.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing 15a15a with 12a+3a12a + 3a: $4a^2 + 12a + 3a + 9$

Step 5: Factor by grouping

Group the terms: $(4a^2 + 12a) + (3a + 9)$

Factor out the GCF from each group:

4a(a+3)+3(a+3)4a(a + 3) + 3(a + 3)

Step 6: Factor out the common factor

Factor out (a+3)(a + 3): $(4a + 3)(a + 3)$

So, the factored form of $4a^2 + 15a + 9$ is $(4a + 3)(a + 3)$.

8) Factoring $20y^2 - y - 1$

Let's tackle one more, this time with 'y': $20y^2 - y - 1$

Step 1: Identify a, b, and c

Here, a=20a = 20, b=βˆ’1b = -1, and c=βˆ’1c = -1.

Step 2: Calculate ac

Multiply aa and cc: 20βˆ—βˆ’1=βˆ’2020 * -1 = -20.

Step 3: Find two numbers that multiply to ac and add up to b

We need two numbers that multiply to -20 and add up to -1. Those numbers are -5 and 4.

Step 4: Rewrite the middle term

Rewrite the original expression, replacing βˆ’y-y with βˆ’5y+4y-5y + 4y: $20y^2 - 5y + 4y - 1$

Step 5: Factor by grouping

Group the terms: $(20y^2 - 5y) + (4y - 1)$

Factor out the GCF from each group:

5y(4yβˆ’1)+1(4yβˆ’1)5y(4y - 1) + 1(4y - 1)

Step 6: Factor out the common factor

Factor out (4yβˆ’1)(4y - 1): $(5y + 1)(4y - 1)$

Therefore, the factored form of $20y^2 - y - 1$ is $(5y + 1)(4y - 1)$.

Key Takeaways and Tips for Factoring

  • Practice makes perfect! The more you factor, the easier it becomes.
  • Always look for a GCF first. If there's a common factor in all terms, factor it out before anything else. This simplifies the problem.
  • Double-check your work. Multiply the factors you found to make sure you get back the original expression.
  • Don't give up! Factoring can be challenging, but with persistence, you'll master it.

Conclusion

Factoring quadratic expressions is a fundamental skill in algebra, and I hope these examples have helped you understand the process better. Remember the key steps: find acac, find the two numbers, rewrite the middle term, factor by grouping, and factor out the common factor. Keep practicing, and you'll become a factoring pro in no time! If you have more questions, ask away! Let’s keep this math discussion going. You've got this!