Learn how to solve for x and find the values that satisfy the equation “for what values of x is x² + 2x + 24 true?” in this comprehensive guide.

## Introduction

When it comes to solving mathematical equations, it’s essential to have a clear understanding of the problem. One such problem is figuring out for what values of x is x² + 2x + 24 true. This equation may look intimidating at first glance, but with the right knowledge and approach, it can be solved effectively. In this article, we’ll walk you through how to solve for x and check the validity of your solution.

## Understanding the problem

Before we dive into solving the equation, let’s take a step back and understand what it means. The equation x² + 2x + 24 is a quadratic equation, which means it’s a polynomial equation of the second degree. It consists of three terms: x², 2x, and 24.

Our goal is to find the values of x that satisfy the equation, or in other words, make it true. This is done by solving for x and finding the roots of the equation. The roots are the values of x that make the equation equal to zero.

To solve for x, we can use the quadratic formula, which is -b ± √(b² – 4ac) / 2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = 2, and c = 24.

## Solving for x

Now that we understand the problem, let’s solve for x. We’ll start by plugging in the values of a, b, and c into the quadratic formula:

x = (-2 ± √(2² – 4(1)(24))) / 2(1)

Simplifying the formula, we get:

x = (-2 ± √(4 – 96)) / 2

x = (-2 ± √-92) / 2

At this point, we have a problem. The square root of a negative number is not a real number, which means there are no real solutions to the equation. This is because the graph of the quadratic equation x² + 2x + 24 is a parabola that opens upwards, and its vertex is above the x-axis. Therefore, the parabola does not intersect the x-axis, and there are no real roots.

However, we can still find the complex roots of the equation. We can rewrite the equation as x² + 2x + 24 = 0, and then complete the square to get:

(x + 1)² + 23 = 0

(x + 1)² = -23

x + 1 = ±√-23

x = -1 ± i√23

So the complex roots of the equation are x = -1 + i√23 and x = -1 – i√23.

## Checking the validity of the solution

Now that we have found the roots of the equation, we need to check the validity of our solution. To do this, we can plug in the values of the roots into the original equation and see if they make it true.

Let’s start with the first root, x = -1 + i√23:

(-1 + i√23)² + 2(-1 + i√23) + 24

= 1 – 2i√23 – 23 + 2 – 2i√23 + 24

= 4

Since the output is not equal to zero, the first root is not valid.

Now, let’s try the second root, x = -1 – i√23:

(-1 – i√23)² + 2(-1 – i√23) + 24

= 1 + 2i√23 – 23 – 2 – 2i√23 + 24

= 4

The output is also 4, which means the second root is valid. Therefore, the answer to the question “for what values of x is x² + 2x + 24 true?” is x = -1 – i√23.

## Checking the validity of the solution

Once you have solved for x, it’s important to check the validity of your solution. One way to do this is by substituting your answer back into the original equation and verifying that it is true.

For example, if you found that x = 4, you would substitute 4 for x in the equation x² + 2x + 24 and simplify it to see if it equals zero.

x² + 2x + 24 = 4² + 2(4) + 24

= 16 + 8 + 24

= 48

Since 48 is not equal to zero, we know that x = 4 is not a valid solution.

## Conclusion

In conclusion, solving for x in the equation x² + 2x + 24 may seem daunting at first, but with the use of the quadratic formula, it can be achieved with ease. It’s important to remember that the roots of the equation are the values of x that make it true, and checking the validity of your solution is crucial to ensure accuracy.

With this knowledge, you can confidently approach solving quadratic equations and tackle more complex mathematical problems. Remember to always take the time to understand the problem and approach it systematically to achieve the best results.