Solving Quadratic Equations X² - 5x - 24 = 0 Using The Quadratic Formula
In the realm of mathematics, quadratic equations hold a significant place, appearing in various contexts and applications. Solving these equations is a fundamental skill, and the quadratic formula stands as a powerful tool for this purpose. This article delves into the process of solving quadratic equations using the quadratic formula, providing a comprehensive guide with explanations, examples, and practical insights.
H2: Understanding Quadratic Equations
At its core, a quadratic equation is a polynomial equation of the second degree. Its general form is expressed as:
ax² + bx + c = 0
where 'a', 'b', and 'c' are constants, and 'x' represents the variable we aim to solve for. The coefficient 'a' cannot be zero, as this would reduce the equation to a linear form. The solutions to a quadratic equation, also known as roots or zeros, are the values of 'x' that satisfy the equation.
Quadratic equations can have up to two distinct real solutions, one real solution (a repeated root), or two complex solutions. The nature of the solutions depends on the discriminant, a value derived from the coefficients of the equation.
H3: The Discriminant
The discriminant, denoted by Δ (Delta), is calculated using the formula:
Δ = b² - 4ac
The discriminant provides valuable information about the nature of the solutions:
- If Δ > 0: The equation has two distinct real solutions.
- If Δ = 0: The equation has one real solution (a repeated root).
- If Δ < 0: The equation has two complex solutions.
Understanding the discriminant allows us to predict the type of solutions we will encounter before applying the quadratic formula.
H2: The Quadratic Formula: A Powerful Solution
The quadratic formula provides a direct method for finding the solutions of any quadratic equation. It is expressed as:
x = (-b ± √(b² - 4ac)) / 2a
This formula elegantly incorporates the coefficients 'a', 'b', and 'c' to determine the solutions for 'x'. The '±' symbol indicates that there are potentially two solutions, one obtained by adding the square root term and the other by subtracting it.
H3: Applying the Quadratic Formula: A Step-by-Step Guide
To effectively utilize the quadratic formula, follow these steps:
- Identify the coefficients: Begin by identifying the values of 'a', 'b', and 'c' in the quadratic equation.
- Calculate the discriminant: Compute the discriminant (Δ = b² - 4ac) to determine the nature of the solutions.
- Substitute into the formula: Substitute the values of 'a', 'b', 'c', and the discriminant into the quadratic formula.
- Simplify: Simplify the expression to obtain the solutions for 'x'. Remember to consider both the '+' and '-' cases from the '±' symbol.
Let's illustrate this process with an example:
Example:
Solve the equation: x² - 5x - 24 = 0
- Identify the coefficients:
- a = 1
- b = -5
- c = -24
- Calculate the discriminant:
- Δ = (-5)² - 4(1)(-24) = 25 + 96 = 121
- Substitute into the formula:
- x = (-(-5) ± √121) / 2(1)
- Simplify:
- x = (5 ± 11) / 2
- x₁ = (5 + 11) / 2 = 8
- x₂ = (5 - 11) / 2 = -3
Therefore, the solutions to the equation x² - 5x - 24 = 0 are x = 8 and x = -3.
H2: Solving x² - 5x - 24 = 0
Let's apply the quadratic formula to the specific equation provided: x² - 5x - 24 = 0. This quadratic equation is in the standard form ax² + bx + c = 0, where:
- a = 1
- b = -5
- c = -24
H3: Step 1: Identify the Coefficients
The first step in solving this quadratic equation using the quadratic formula is to correctly identify the coefficients a, b, and c. In the given equation, x² - 5x - 24 = 0, we have:
- a is the coefficient of x², which is 1.
- b is the coefficient of x, which is -5.
- c is the constant term, which is -24.
These values are crucial for the correct application of the quadratic formula.
H3: Step 2: Calculate the Discriminant
Before plugging the coefficients into the quadratic formula, it's beneficial to calculate the discriminant. The discriminant, denoted as Δ, is given by the formula Δ = b² - 4ac. This value tells us about the nature of the roots (solutions) of the quadratic equation:
- If Δ > 0, the equation has two distinct real roots.
- If Δ = 0, the equation has one real root (a repeated root).
- If Δ < 0, the equation has two complex roots.
In our case, the discriminant is:
Δ = (-5)² - 4(1)(-24) = 25 + 96 = 121
Since Δ = 121 > 0, we know that the equation has two distinct real roots.
H3: Step 3: Apply the Quadratic Formula
The quadratic formula is a general formula that provides the solutions to any quadratic equation of the form ax² + bx + c = 0. The formula is:
x = (-b ± √(b² - 4ac)) / 2a
Now, we substitute the values of a, b, and c into the formula:
x = (-(-5) ± √((-5)² - 4(1)(-24))) / (2(1))
H3: Step 4: Simplify the Expression
Next, we simplify the expression step by step:
x = (5 ± √(25 + 96)) / 2 x = (5 ± √121) / 2 x = (5 ± 11) / 2
This gives us two possible solutions for x:
- x₁ = (5 + 11) / 2 = 16 / 2 = 8
- x₂ = (5 - 11) / 2 = -6 / 2 = -3
Therefore, the solutions to the quadratic equation x² - 5x - 24 = 0 are x = 8 and x = -3.
H3: Step 5: Enter the Smallest Solution First
The problem asks to enter the smallest solution first. Comparing the two solutions, -3 is smaller than 8. So, the smallest solution is -3, and the other solution is 8.
H2: Practical Applications of Quadratic Equations
Quadratic equations are not merely abstract mathematical concepts; they find widespread applications in various fields, including:
- Physics: Projectile motion, where the trajectory of an object through the air can be modeled using a quadratic equation.
- Engineering: Designing structures, calculating stress and strain, and optimizing system performance often involve quadratic equations.
- Finance: Calculating compound interest, modeling investment growth, and determining loan payments rely on quadratic relationships.
- Computer Graphics: Quadratic equations are used to create curves and surfaces in computer graphics and animation.
The ability to solve quadratic equations empowers us to tackle real-world problems across diverse disciplines.
H2: Conclusion: Mastering the Quadratic Formula
The quadratic formula is an indispensable tool for solving quadratic equations. Its power lies in its ability to provide solutions for any quadratic equation, regardless of the complexity of the coefficients. By understanding the steps involved in applying the formula and appreciating its practical applications, we can confidently navigate the world of quadratic equations and their significance in various fields.
This article has provided a comprehensive guide to solving quadratic equations using the quadratic formula, equipping you with the knowledge and skills to tackle these equations with confidence. Remember to practice regularly and explore the diverse applications of quadratic equations to further solidify your understanding.
