Solving P^2 - 10p - 11 = 0 By Factoring A Step-by-Step Guide

by Chloe Fitzgerald 61 views

Hey there, math enthusiasts! Today, we're going to dive into solving a quadratic equation by factoring. Factoring is a super handy technique for finding the solutions (also called roots or zeros) of a quadratic equation. Let's break down the equation p^2 - 10p - 11 = 0 step by step. So, grab your pencils, and let's get started!

Understanding Quadratic Equations

Before we jump into factoring, let's quickly recap what a quadratic equation is. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. In our case, the equation is p^2 - 10p - 11 = 0, so we can see that a = 1, b = -10, and c = -11. The goal is to find the values of 'p' that make the equation true. These values are the solutions to the equation.

Why Factoring?

Factoring is one of the most straightforward methods for solving quadratic equations, especially when the equation can be easily factored. It involves breaking down the quadratic expression into a product of two binomials. When we set each binomial equal to zero, we can solve for 'p'. This method relies on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Factoring is a classic method that provides a solid foundation for more advanced algebraic techniques. It’s also super satisfying when you can quickly spot the factors and solve the equation in a few steps!

Steps to Factoring

Here’s the general approach we'll use:

  1. Identify the coefficients: In our equation p^2 - 10p - 11 = 0, we've already identified that a = 1, b = -10, and c = -11.
  2. Find two numbers: We need to find two numbers that multiply to 'c' (which is -11) and add up to 'b' (which is -10). This is the key step in factoring, and it might take a bit of trial and error. But don't worry, we'll walk through it.
  3. Write the factored form: Once we find those two numbers, we can write the quadratic expression in factored form as (p + number 1)(p + number 2) = 0.
  4. Set each factor to zero: Using the zero-product property, we set each factor equal to zero and solve for 'p'.
  5. Solve for p: This will give us the solutions to the quadratic equation.

Factoring p^2 - 10p - 11 = 0

Alright, let's apply these steps to our equation p^2 - 10p - 11 = 0.

Step 1: Identify the Coefficients

As we've already mentioned, our coefficients are:

  • a = 1
  • b = -10
  • c = -11

Step 2: Find Two Numbers

This is the trickiest part, but we can totally do it! We need to find two numbers that:

  • Multiply to -11 (the value of 'c')
  • Add up to -10 (the value of 'b')

Let's think about the factors of -11. Since 11 is a prime number, its only factors are 1 and 11. To get a product of -11, one of the factors must be negative. So, our pairs of factors could be:

  • 1 and -11
  • -1 and 11

Now, let's check which pair adds up to -10:

  • 1 + (-11) = -10 This is it!
  • -1 + 11 = 10

So, the two numbers we're looking for are 1 and -11.

Step 3: Write the Factored Form

Now that we have our two numbers, 1 and -11, we can write the quadratic expression in factored form:

(p + 1)(p - 11) = 0

See how we used the numbers we found directly in the binomials? The +1 comes from the number 1, and the -11 comes from the number -11. Factoring is like reverse-FOILing (First, Outer, Inner, Last), so if you want to double-check, you can multiply these two binomials together to make sure you get back to the original equation.

Step 4: Set Each Factor to Zero

Using the zero-product property, we set each factor equal to zero:

  • p + 1 = 0
  • p - 11 = 0

This step is crucial because it allows us to split the original quadratic equation into two simpler linear equations. Solving linear equations is much easier, right?

Step 5: Solve for p

Now, let's solve each equation for 'p':

  1. For p + 1 = 0, subtract 1 from both sides:

    p = -1

  2. For p - 11 = 0, add 11 to both sides:

    p = 11

And there you have it! We've found the solutions to the quadratic equation. The values of 'p' that satisfy the equation are -1 and 11.

Expressing the Solution

When there's more than one solution, we typically separate the values with a comma. So, the solutions to the equation p^2 - 10p - 11 = 0 are:

p = -1, 11

These are the points where the parabola represented by the quadratic equation crosses the x-axis. Understanding these solutions is fundamental in various mathematical and real-world applications. From physics to engineering, quadratic equations pop up everywhere, making it super important to know how to solve them.

Checking Our Work

It's always a good idea to check our solutions to make sure they're correct. We can do this by plugging each value of 'p' back into the original equation and seeing if it holds true.

Checking p = -1

Substitute p = -1 into the equation p^2 - 10p - 11 = 0:

(-1)^2 - 10(-1) - 11 = 0

1 + 10 - 11 = 0

11 - 11 = 0

0 = 0 This checks out!

Checking p = 11

Substitute p = 11 into the equation p^2 - 10p - 11 = 0:

(11)^2 - 10(11) - 11 = 0

121 - 110 - 11 = 0

121 - 121 = 0

0 = 0 This also checks out!

Since both solutions satisfy the original equation, we can be confident that our factoring was correct.

Alternative Methods for Solving Quadratic Equations

While factoring is a fantastic method, it's not always the easiest or most practical approach, especially when the equation is more complex. Let's briefly touch on a couple of alternative methods for solving quadratic equations:

1. Quadratic Formula

The quadratic formula is a universal method that can be used to solve any quadratic equation, regardless of whether it can be easily factored. The formula is:

p = (-b ± √(b^2 - 4ac)) / (2a)

Where 'a', 'b', and 'c' are the coefficients of the quadratic equation ax^2 + bx + c = 0. In our case, a = 1, b = -10, and c = -11. Plugging these values into the formula will also give us the solutions p = -1 and p = 11. The quadratic formula is a reliable tool in your math arsenal, especially when factoring becomes tricky.

2. Completing the Square

Completing the square is another method that involves transforming the quadratic equation into a perfect square trinomial. This method is particularly useful for understanding the structure of quadratic equations and for deriving the quadratic formula itself. While it can be a bit more involved than factoring, completing the square provides valuable insights into the nature of quadratic equations. It's like understanding the inner workings of a clock rather than just reading the time!

Tips and Tricks for Factoring

Factoring can sometimes feel like a puzzle, but with practice, you'll become a pro! Here are a few tips and tricks to help you along the way:

  • Always look for a common factor first: Before diving into factoring the quadratic expression, check if there's a common factor that can be factored out from all the terms. This simplifies the equation and makes it easier to factor.
  • Practice, practice, practice: The more you practice factoring, the quicker and more confident you'll become. Try factoring different types of quadratic equations to build your skills.
  • Use the reverse-FOIL method: As mentioned earlier, factoring is the reverse of the FOIL (First, Outer, Inner, Last) method used for multiplying binomials. Keep this in mind when finding the two numbers that multiply to 'c' and add up to 'b'.
  • Don't give up: Some quadratic equations are trickier to factor than others. If you're stuck, take a break, try a different approach, or use an alternative method like the quadratic formula.

Conclusion

So, guys, we've successfully solved the quadratic equation p^2 - 10p - 11 = 0 by factoring! We found that the solutions are p = -1 and p = 11. Remember, factoring is a powerful technique for solving quadratic equations, and it's a fundamental skill in algebra. Keep practicing, and you'll become a factoring whiz in no time!

We also touched on alternative methods like the quadratic formula and completing the square, which are valuable tools for solving quadratic equations that might be harder to factor. Keep these methods in your toolkit, and you'll be well-equipped to tackle any quadratic equation that comes your way. Happy solving!