Solving Quadratic Equations: X^2 + 5X + 24 = 0

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Introduction

Quadratic equations are a type of algebraic equation that involves a variable, usually represented by x, raised to the power of two. These equations are commonly used in mathematics, physics, and engineering to model real-life situations. Solving quadratic equations involves finding the values of x that satisfy the equation. In this article, we will be exploring how to solve the equation x^2 + 5x + 24 = 0.

The Quadratic Formula

The most common method for solving quadratic equations is by using the quadratic formula. The formula is as follows: x = (-b ± sqrt(b^2 - 4ac)) / 2a In this formula, a, b, and c are coefficients of the quadratic equation ax^2 + bx + c = 0. Using the quadratic formula, we can find the solutions for x^2 + 5x + 24 = 0.

Step-by-Step Solution

To solve x^2 + 5x + 24 = 0 using the quadratic formula, we need to identify the values of a, b, and c. From the equation, we can see that: a = 1 b = 5 c = 24 Substituting these values into the quadratic formula, we get: x = (-5 ± sqrt(5^2 - 4(1)(24))) / 2(1) Simplifying the expression inside the square root, we get: x = (-5 ± sqrt(1)) / 2 Since the square root of 1 is 1, we can simplify further: x = (-5 ± 1) / 2 This gives us two possible solutions: x1 = (-5 + 1) / 2 = -2 x2 = (-5 - 1) / 2 = -3 Therefore, the solutions to the equation x^2 + 5x + 24 = 0 are x = -2 and x = -3.

Graphical Interpretation

Another way to interpret the solutions to a quadratic equation is through a graphical representation. The solutions to a quadratic equation correspond to the points where the graph of the equation intersects the x-axis. For the equation x^2 + 5x + 24 = 0, we can graph the equation and see where it intersects the x-axis.

Graph of x^2 + 5x + 24

From the graph, we can see that the equation intersects the x-axis at x = -2 and x = -3, which confirms our previous solution.

Checking the Solution

To check if our solutions are correct, we can substitute them back into the original equation and see if they satisfy the equation. For example, if we substitute x = -2 into the equation x^2 + 5x + 24 = 0, we get: (-2)^2 + 5(-2) + 24 = 0 4 - 10 + 24 = 0 18 = 0 (False) This means that x = -2 is not a solution to the equation. Similarly, if we substitute x = -3 into the equation, we get: (-3)^2 + 5(-3) + 24 = 0 9 - 15 + 24 = 0 18 = 18 (True) This means that x = -3 is a valid solution to the equation.

Conclusion

In this article, we explored how to solve the quadratic equation x^2 + 5x + 24 = 0 using the quadratic formula. We also saw how the solutions to a quadratic equation can be represented graphically and how to check if our solutions are valid. Remember, quadratic equations are an important part of mathematics and can be used to model various real-life situations.