Introduction
Solving quadratic equations may seem daunting at first, but fear not! In this step-by-step guide, we will demystify the process of solving the quadratic equation 4x ^ 2 – 5x – 12 = 0. By the end of this article, you’ll not only have a clear understanding of the solution but also gain confidence in your mathematical abilities. Let’s dive right in.
Understanding Quadratic Equations
Before we delve into the solution, let’s understand what a quadratic equation is. A quadratic equation is a second-degree polynomial equation in a single variable, usually written in the form ax^2 + bx + c = 0, where ‘a,’ ‘b,’ and ‘c’ are constants.
Solving the Quadratic Equation 4x ^ 2 – 5x – 12 = 0
Now, let’s focus on our target equation: 4x ^ 2 – 5x – 12 = 0. We’ll break down the solution into manageable steps.
Step 1: Identify the Coefficients
To begin, identify the coefficients ‘a,’ ‘b,’ and ‘c’ in the equation. In our case, ‘a’ is 4, ‘b’ is -5, and ‘c’ is -12.
Step 2: Calculate the Discriminant
The discriminant, denoted by ‘D,’ is a crucial factor in solving quadratic equations. It is calculated using the formula D = b^2 – 4ac. For our equation, D = (-5)^2 – 4(4)(-12).
Step 3: Determine the Nature of Roots
Based on the discriminant’s value, you can determine the nature of the roots:
- If D > 0, there are two distinct real roots.
- If D = 0, there is one real root.
- If D < 0, there are two complex roots.
Step 4: Calculate the Roots
Now, let’s calculate the roots using the quadratic formula: x = (-b ± √D) / (2a). Plug in the values of ‘a,’ ‘b,’ and ‘c’ along with the discriminant to find the roots.
Solving the Quadratic Equation 4x ^ 2 – 5x – 12 = 0 using Factoring
Factoring is a technique that involves breaking down a quadratic equation into simpler expressions, making it easier to find the roots. Let’s see how it applies to our equation:
Step 1: Write the equation in the form ax^2 + bx + c = 0. In our case, a = 4, b = -5, and c = -12.
Step 2: Look for two numbers that multiply to ‘a * c’ (product of ‘a’ and ‘c’) and add up to ‘b.’ For our equation, ‘a * c’ = (4 * -12) = -48. We need two numbers that multiply to -48 and add up to -5. These numbers are -8 and 6.
Step 3: Rewrite the middle term (-5x) using the numbers from Step 2: 4x ^ 2 – 5x – 12 = 0
Step 4: Group the terms and factor by grouping: (4x^2 – 8x) + (6x – 12) = 0 4x(x – 2) + 6(x – 2) = 0
Step 5: Factor out the common factor, which is (x – 2): (x – 2)(4x + 6) = 0
Step 6: Set each factor equal to zero and solve for ‘x’: x – 2 = 0 and 4x + 6 = 0 x = 2 4x = -6 x = -6/4 x = -3/2
So, the solutions to the equation 4x^2 – 5x – 12 = 0 using factoring are x = 2 and x = -3/2.
Solving the Quadratic Equation 4x ^ 2 – 5x – 12 = 0 using the Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations of the form ax^2 + bx + c = 0. It involves the formula:
x = (-b ± √(b^2 – 4ac)) / (2a)
Let’s apply the quadratic formula to our equation:
Step 1: Identify the coefficients: a = 4, b = -5, and c = -12.
Step 2: Plug the coefficients into the quadratic formula: x = (-(-5) ± √((-5)^2 – 4(4)(-12))) / (2(4))
Step 3: Simplify and solve: x = (5 ± √(25 + 192)) / 8 x = (5 ± √217) / 8
So, the solutions to the equation 4x ^ 2 – 5x – 12 = 0 using the quadratic formula are x = (5 + √217) / 8 and x = (5 – √217) / 8.
Comparing Factoring and the Quadratic Formula
Now that we’ve solved the equation using both methods, let’s compare them:
Advantages of Factoring:
- Factoring can be quicker and more straightforward when the equation is easily factorable.
- It provides insight into how the equation can be simplified.
- It may result in whole number solutions, which can be easier to work with.
Advantages of the Quadratic Formula:
- The quadratic formula works for all quadratic equations, even those that aren’t easily factorable.
- It provides both real and complex solutions.
- It’s a reliable method when you’re unsure if factoring is possible.
FAQs
How do I know if a quadratic equation has real roots?
If the discriminant (D) is greater than or equal to zero, the quadratic equation has real roots.
Can I use the quadratic formula for any quadratic equation?
Yes, the quadratic formula is a universal method for solving quadratic equations of the form ax^2 + bx + c = 0.
What if the discriminant is negative?
If the discriminant is negative, the quadratic equation has two complex roots, which are conjugates of each other.
Is there any shortcut for solving quadratic equations?
While the quadratic formula is the most reliable method, some quadratic equations can be factored for quicker solutions.
Can I use an online calculator to solve quadratic equations?
Yes, there are many online quadratic equation solvers available, but understanding the manual method is essential for learning.
Are quadratic equations used in real-life scenarios?
Yes, quadratic equations have practical applications in physics, engineering, economics, and various other fields.
Conclusion
Congratulations! You’ve successfully learned how to solve the quadratic equation 4x ^ 2 – 5x – 12 = 0 step-by-step. Remember that practice makes perfect, and the more you work with quadratic equations, the more confident you’ll become. Embrace the world of mathematics and explore its fascinating applications.
Both factoring and the quadratic formula are valuable tools for solving quadratic equations like 4x ^ 2 – 5x – 12 = 0. The choice between them depends on the complexity of the equation and your comfort level with each method. Whether you prefer the elegance of factoring or the universality of the quadratic formula, you now have the knowledge to tackle quadratic equations with confidence.
