Solving (x-2)(x+5)=18: Find The Right Solution!

Hey guys! Let's dive into solving a quadratic equation. We've got a fun one today: $(x-2)(x+5)=18$. We need to figure out which of the given options is the correct solution. Let's break it down step-by-step so it's super easy to follow.

Understanding the Problem

So, the million-dollar question is: Which value of x makes the equation $(x-2)(x+5)=18$ true? We have a few options to test:

• A. $x=-10$
• B. $x=-7$
• C. $x=-4$
• D. $x=-2$

Before we start plugging in numbers, let's remember what it means to solve an equation. We are looking for the value of x that, when substituted into the equation, makes both sides equal. In this case, we want the left side, $(x-2)(x+5)$, to equal 18.

Method 1: Expanding and Solving the Quadratic Equation

One way to solve this is by expanding the left side of the equation and rearranging it into the standard quadratic form, which is $ax^2 + bx + c = 0$. Then, we can either factor the quadratic or use the quadratic formula to find the solutions.

Let's expand $(x-2)(x+5)$:

$(x-2)(x+5) = x(x+5) - 2(x+5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10$

Now, our equation looks like this:

$x^2 + 3x - 10 = 18$

To get it into the standard form, we subtract 18 from both sides:

$x^2 + 3x - 10 - 18 = 0$

$x^2 + 3x - 28 = 0$

Now we need to factor this quadratic. We're looking for two numbers that multiply to -28 and add up to 3. Those numbers are 7 and -4. So, we can factor the quadratic as follows:

$(x+7)(x-4) = 0$

Setting each factor equal to zero gives us the possible solutions for x:

$x+7 = 0$ or $x-4 = 0$

Solving for x, we get:

$x = -7$ or $x = 4$

So, the solutions to the equation are $x = -7$ and $x = 4$. Now, let's check our options.

Method 2: Testing the Options

Since we have multiple-choice options, we can plug each value of x into the original equation $(x-2)(x+5)=18$ and see which one makes the equation true. This can be a quicker method if the solutions are integers.

Let's test each option:

• A. $x = -10$

  $(-10-2)(-10+5) = (-12)(-5) = 60$. This is not equal to 18, so $x=-10$ is not a solution.

• B. $x = -7$

  $(-7-2)(-7+5) = (-9)(-2) = 18$. This is equal to 18, so $x=-7$ is a solution.

• C. $x = -4$

  $(-4-2)(-4+5) = (-6)(1) = -6$. This is not equal to 18, so $x=-4$ is not a solution.

• D. $x = -2$

  $(-2-2)(-2+5) = (-4)(3) = -12$. This is not equal to 18, so $x=-2$ is not a solution.

From testing the options, we found that only $x=-7$ satisfies the equation.

The Correct Answer

Based on both methods, the correct solution is:

B. $x = -7$

Why This Matters

Understanding how to solve quadratic equations is super important in math. You'll see these types of problems pop up everywhere, from simple algebra to more complex calculus. Plus, being able to solve equations quickly can help you nail those timed tests!

Tips for Solving Quadratic Equations

Here are some quick tips to keep in mind when you're tackling quadratic equations:

• Always rearrange the equation into the standard form $ax^2 + bx + c = 0$ before trying to factor or use the quadratic formula.
• Look for easy factors. Factoring can save you a lot of time if you can spot the right numbers quickly.
• Don't forget the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. It's a lifesaver when factoring doesn't work.
• Check your answers! Plug your solutions back into the original equation to make sure they work.
• Practice makes perfect. The more you solve these equations, the easier they'll become.

Common Mistakes to Avoid

• Forgetting to distribute: When expanding expressions like $(x-2)(x+5)$, make sure you multiply each term correctly.
• Incorrectly factoring: Double-check your factors to make sure they multiply back to the original quadratic.
• Sign errors: Pay close attention to signs when rearranging equations or using the quadratic formula. A simple sign mistake can throw off your entire answer.
• Only finding one solution: Quadratic equations usually have two solutions. Make sure you find both of them.

Real-World Applications

You might be wondering, "Where am I ever going to use this in real life?" Well, quadratic equations show up in many fields, including:

• Physics: Calculating the trajectory of a projectile.
• Engineering: Designing structures and optimizing performance.
• Economics: Modeling supply and demand curves.
• Computer Science: Creating algorithms and simulations.

So, even if it doesn't seem like it now, understanding quadratic equations can open up a lot of doors!

Conclusion

Alright, guys, we've walked through how to solve the equation $(x-2)(x+5)=18$ and found that the correct solution is B. $x = -7$. Whether you prefer expanding and factoring or testing the options, remember to take your time, double-check your work, and don't be afraid to ask for help if you get stuck. Keep practicing, and you'll be a quadratic equation pro in no time!