Rewriting $x^2 + X - 72$ A Comprehensive Guide

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Hey guys! Let's dive into the fascinating world of quadratic expressions and learn how to rewrite them effectively. In this article, we're going to tackle the expression $x^2 + x - 72$ and explore different ways to manipulate it. Whether you're a student brushing up on algebra or just someone who loves mathematical puzzles, this guide is for you. We'll break down the steps, explain the concepts, and provide plenty of examples to make sure you've got a solid grasp on the topic. So, let's get started!

Understanding Quadratic Expressions

Before we jump into rewriting $x^2 + x - 72$, let's take a moment to understand what quadratic expressions are all about. At its core, a quadratic expression is a polynomial expression of degree two. This means the highest power of the variable (in our case, x) is 2. The general form of a quadratic expression is: $ax^2 + bx + c$, where a, b, and c are constants, and a is not equal to 0. Why can't a be 0? Well, if a were 0, the $x^2$ term would vanish, and we'd be left with a linear expression ($bx + c$) instead of a quadratic one.

In our specific expression, $x^2 + x - 72$, we can identify the coefficients as follows:

• a = 1 (the coefficient of $x^2$)
• b = 1 (the coefficient of x)
• c = -72 (the constant term)

Understanding these coefficients is crucial because they dictate the behavior and properties of the quadratic expression. For instance, the sign of a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The values of b and c influence the position and shape of the parabola.

Quadratic expressions are everywhere in mathematics and real-world applications. They pop up in physics (think projectile motion), engineering (designing bridges and structures), and even economics (modeling costs and revenues). Being able to manipulate and rewrite these expressions is a fundamental skill that opens doors to solving a wide range of problems. Whether you're trying to find the roots of an equation, sketch a graph, or optimize a system, a solid understanding of quadratic expressions is your best friend.

So, now that we've got the basics down, let's move on to the exciting part: how to rewrite $x^2 + x - 72$. We'll explore different techniques, starting with the most common method: factoring. Buckle up, because we're about to turn this expression inside out and see what makes it tick!

Factoring the Quadratic Expression

Alright, let's get our hands dirty and factor the quadratic expression $x^2 + x - 72$. Factoring is like reverse multiplication – we're trying to find two binomials (expressions with two terms) that, when multiplied together, give us our original quadratic expression. This method is super useful for solving quadratic equations, simplifying expressions, and understanding the roots of the quadratic.

The general idea behind factoring a quadratic expression of the form $ax^2 + bx + c$ is to find two numbers (let's call them p and q) such that:

• p + q = b (the coefficient of x)
• p * q* = a * c* (the product of the coefficient of $x^2$ and the constant term)

In our case, $x^2 + x - 72$, we have:

• a = 1
• b = 1
• c = -72

So, we need to find two numbers p and q such that:

• p + q = 1
• p * q* = 1 * (-72) = -72

This is where the fun begins! We need to think about pairs of numbers that multiply to -72. Since the product is negative, we know one number must be positive and the other negative. Let's list some pairs:

• 1 and -72
• -1 and 72
• 2 and -36
• -2 and 36
• 3 and -24
• -3 and 24
• 4 and -18
• -4 and 18
• 6 and -12
• -6 and 12
• 8 and -9
• -8 and 9

Now, we need to find the pair that adds up to 1. Looking at our list, we can see that -8 and 9 fit the bill:

• -8 + 9 = 1
• -8 * 9 = -72

Awesome! We've found our numbers. Now we can rewrite the middle term (x) using these numbers:

$x^2 + x - 72 = x^2 - 8x + 9x - 72$

Next, we'll use a technique called factoring by grouping. We'll group the first two terms and the last two terms:

$(x^2 - 8x) + (9x - 72)$

Now, we'll factor out the greatest common factor (GCF) from each group:

$x(x - 8) + 9(x - 8)$

Notice that we have a common factor of (x - 8) in both terms. We can factor this out:

$(x - 8)(x + 9)$

Boom! We've factored the quadratic expression. So, $x^2 + x - 72$ can be rewritten as $(x - 8)(x + 9)$. This is super helpful because it gives us insights into the roots of the equation $x^2 + x - 72 = 0$. The roots are the values of x that make the expression equal to zero, which are x = 8 and x = -9. Understanding factoring is a cornerstone of algebra, and it's going to be a valuable tool in your math toolkit.

Completing the Square

Let's explore another powerful technique for rewriting quadratic expressions: completing the square. This method is a bit more involved than factoring, but it's incredibly versatile and useful for solving quadratic equations, finding the vertex of a parabola, and even integrating certain functions in calculus. Completing the square essentially transforms a quadratic expression into a perfect square trinomial, plus or minus a constant term. What's a perfect square trinomial, you ask? It's a trinomial that can be factored into the form $(x + p)^2$ or $(x - p)^2$.

The general idea behind completing the square for a quadratic expression of the form $ax^2 + bx + c$ is to manipulate the expression so that it looks like $a(x + h)^2 + k$, where h and k are constants. Let's break down the steps for our expression, $x^2 + x - 72$.

Step 1: Ensure the coefficient of $x^2$ is 1.

In our case, the coefficient of $x^2$ is already 1, so we can skip this step. If it weren't 1, we'd need to factor out the coefficient from the $x^2$ and x terms.

Step 2: Move the constant term to the right side (if we were solving an equation).

Since we're just rewriting the expression, we'll keep the constant term on the left side for now.

Step 3: Take half of the coefficient of x, square it, and add it to both sides.

The coefficient of x in our expression is 1. Half of 1 is 1/2, and squaring that gives us (1/2)^2 = 1/4. So, we'll add 1/4 to our expression:

$x^2 + x + 1/4 - 72$

Step 4: Rewrite the first three terms as a perfect square trinomial.

The first three terms, $x^2 + x + 1/4$, form a perfect square trinomial. We can rewrite it as:

$(x + 1/2)^2$

Why does this work? Remember, a perfect square trinomial is of the form $(x + p)^2 = x^2 + 2px + p^2$. In our case, 2p = 1, so p = 1/2, and $p^2$ = 1/4. This is why we added 1/4 – to complete the square.

Step 5: Simplify the expression.

Now we need to combine the constant terms:

$(x + 1/2)^2 - 72 - 1/4$

To subtract the fractions, we need a common denominator. Let's convert 72 to a fraction with a denominator of 4:

72 = 72/1 = (72 * 4) / (1 * 4) = 288/4

So, our expression becomes:

$(x + 1/2)^2 - 288/4 - 1/4$

$(x + 1/2)^2 - 289/4$

There you have it! We've rewritten $x^2 + x - 72$ by completing the square as $(x + 1/2)^2 - 289/4$. This form is incredibly useful for finding the vertex of the parabola represented by the quadratic expression. The vertex is at the point (-1/2, -289/4). Completing the square might seem a bit tricky at first, but with practice, it becomes a valuable tool in your mathematical arsenal. It provides a different perspective on quadratic expressions and helps you unlock their hidden properties.

Using the Quadratic Formula

Now, let's talk about another powerful tool for dealing with quadratic expressions: the quadratic formula. This formula is like the Swiss Army knife of quadratic equations – it can solve any quadratic equation, regardless of whether it can be factored easily or not. The quadratic formula provides a direct way to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$.

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula might look a bit intimidating at first, but don't worry! We'll break it down and see how it works for our expression. Remember, in our expression, $x^2 + x - 72$, we have:

• a = 1
• b = 1
• c = -72

Let's plug these values into the quadratic formula:

$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-72)}}{2(1)}$

Now, let's simplify step by step:

$x = \frac{-1 \pm \sqrt{1 + 288}}{2}$

$x = \frac{-1 \pm \sqrt{289}}{2}$

The square root of 289 is 17, so we have:

$x = \frac{-1 \pm 17}{2}$

This gives us two possible solutions for x:

1. $x = \frac{-1 + 17}{2} = \frac{16}{2} = 8$
2. $x = \frac{-1 - 17}{2} = \frac{-18}{2} = -9$

Hey, look at that! We got the same roots (8 and -9) that we found by factoring. This is a great way to verify our previous result and shows the consistency of different methods in mathematics. The quadratic formula is especially useful when dealing with quadratic expressions that are difficult or impossible to factor. It provides a reliable way to find the roots, which are essential for understanding the behavior of the quadratic function.

Graphing the Quadratic Expression

Visualizing a quadratic expression can give us a deeper understanding of its properties. Graphing a quadratic expression of the form $ax^2 + bx + c$ results in a parabola, a U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola is the point where the curve changes direction, and it represents the minimum or maximum value of the quadratic expression.

Let's consider our expression, $x^2 + x - 72$. We know that a = 1, which is positive, so the parabola opens upwards. This means the vertex will be the minimum point on the curve.

To graph this expression, we can follow these steps:

1. Find the vertex: We already found the x-coordinate of the vertex when we completed the square. It's -1/2. To find the y-coordinate, we plug this value back into the original expression:

   $y = (-1/2)^2 + (-1/2) - 72$

   $y = 1/4 - 1/2 - 72$

   $y = 1/4 - 2/4 - 288/4$

   $y = -289/4$

   So, the vertex is at the point (-1/2, -289/4), which is approximately (-0.5, -72.25).

2. Find the roots (x-intercepts): We already found the roots by factoring and using the quadratic formula. They are x = 8 and x = -9. These are the points where the parabola intersects the x-axis.

3. Find the y-intercept: The y-intercept is the point where the parabola intersects the y-axis. To find it, we set x = 0 in the original expression:

   $y = (0)^2 + (0) - 72$

   $y = -72$

   So, the y-intercept is at the point (0, -72).

4. Plot the points and sketch the parabola: Now we have the vertex, the roots, and the y-intercept. We can plot these points on a graph and sketch a smooth U-shaped curve that passes through these points. The parabola will be symmetric about the vertical line that passes through the vertex (the axis of symmetry).

Graphing the quadratic expression gives us a visual representation of its behavior. We can see the minimum value, the points where the expression equals zero, and how the expression changes as x changes. This visual understanding is a powerful complement to the algebraic techniques we've explored. It's like having a map to guide us through the world of quadratic expressions.

Conclusion: Mastering Quadratic Expressions

Wow, we've covered a lot of ground in this guide! We started with the basics of quadratic expressions, learned how to factor them, completed the square, used the quadratic formula, and even graphed them. These are all essential skills for anyone working with algebra and beyond. Mastering these techniques not only helps you solve equations but also gives you a deeper understanding of the underlying mathematical concepts.

Quadratic expressions are everywhere, from physics to engineering to economics. Being able to manipulate and rewrite them is a valuable skill that opens doors to solving a wide range of problems. Whether you're trying to find the roots of an equation, sketch a graph, or optimize a system, a solid understanding of quadratic expressions is your best friend.

Remember, the key to mastering any mathematical concept is practice. So, keep working on different examples, try different techniques, and don't be afraid to make mistakes. Mistakes are just learning opportunities in disguise. The more you practice, the more comfortable and confident you'll become with quadratic expressions.

So, guys, keep exploring the world of mathematics, keep asking questions, and keep learning. You've got this! And remember, quadratic expressions might seem complicated at first, but with a little effort and the right tools, you can conquer them all. Happy math-ing!