Quadratic equations form a fundamental part of algebra and have significant applications in various fields including physics, engineering, and economics. In this article, we will discuss on how to solve the quadratic equation 4x^2 – 5x – 12 = 0 and discuss what exactly are Quadratic equations and different methods to solve Quadratic equations.
Contents
What are Quadratic Equations?
Quadratic equations are math problems where the variable is squared (raised to the power of two). They are very important in algebra and are used in subjects like physics, engineering, and economics. Typical the Quadratic Equations looks like ax^2 + bx + c = 0 where a, b, and c are constants, and x is the variable.
Here in this article we will explain different methods to solve Quadratic Equations and also explain how to solve 4x^2 – 5x – 12 = 0.
For this specific equation 4x^2 – 5x – 12 = 0:
a = 4, b = -5, c = -12
Methods to Solve the Quadratic Equation 4x^2 – 5x – 12 = 0
There are several methods to solve quadratic equations, including:
1. Factoring
2. Completing the Square
3. Using the Quadratic Formula
4. Graphical Method
Let’s apply these methods to solve 4x^2 – 5x – 12 = 0.
1. Factoring Method to Solve 4x^2 – 5x – 12 = 0
Factoring involves expressing the quadratic equation as a product of two binomials. However, not all quadratic equations can be factored easily. Let’s see if this method applies to our equation.
We look for two numbers that multiply to a * c = 4 * -12 = -48 and add to b = -5. These numbers are -8 and 6.
Thus, we rewrite the middle term -5x using -8 and 6:
4x^2 – 8x + 6x – 12 = 0
Next, we factor by grouping:
4x(x – 2) + 6(x – 2) = 0
(4x + 6)(x – 2) = 0
This gives us the factors (4x + 6) and (x – 2). Setting each factor to zero provides the solutions:
4x + 6 = 0 => x = -3/2
x – 2 = 0 => x = 2
2. Completing the Square Method to Solve 4x^2 – 5x – 12 = 0
Completing the square involves rewriting the quadratic equation in the form (x – p)^2 = q. Here’s the process:
Start with the original equation:
4x^2 – 5x – 12 = 0
Divide through by 4 to simplify:
x^2 – 5/4x – 3 = 0
Move the constant term to the right side:
x^2 – 5/4x = 3
Add the square of half the coefficient of x to both sides:
x^2 – 5/4x + (5/8)^2 = 3 + (5/8)^2
x^2 – 5/4x + 25/64 = 3 + 25/64
(x – 5/8)^2 = 217/64
Take the square root of both sides:
x – 5/8 = ±√(217/64)
x = 5/8 ± √(217)/8
Simplifying further:
x = (5 ± √(217))/8
Therefore, x can be either 2.466 or −1.216-.
These are the roots of the equation, although in this particular case, completing the square leads to a more complex expression.
3. Quadratic Formula Method to Solve 4x^2 – 5x – 12 = 0
The quadratic formula is a universal method to solve any quadratic equation:
x = (-b ± √(b^2 – 4ac)) / 2a
For our equation 4x^2 – 5x – 12 = 0:
a = 4, b = -5, c = -12
Plugging these values into the formula:
x = (5 ± √(25 + 192)) / 8
x = (5 ± √(217)) / 8
Thus, the solutions are:
x = (5 + √(217)) / 8
x = (5 – √(217)) / 8.
Therefore, x can be either 2.466 or −1.216-.
4. Graphical Method to Solve 4x^2 – 5x – 12 = 0
Graphing the quadratic equation provides a visual representation of its solutions. The graph of y = 4x^2 – 5x – 12 is a parabola. The x-intercepts of this parabola correspond to the solutions of the equation 4x^2 – 5x – 12 = 0.
To sketch the graph:
1. Identify the vertex using x = -b/2a:
x = 5/8
2. Calculate y at the vertex:
y = 4(5/8)^2 – 5(5/8) – 12
y = 4(25/64) – 25/8 – 12
y = 100/64 – 200/64 – 768/64
y = (100 – 200 – 768) / 64 = -13.5625
3. Plot the vertex (5/8, -13.5625) and the x-intercepts x = -3/2 and x = 2.
The parabola opens upwards (since a = 4 > 0) and crosses the x-axis at the solutions we found.
Conclusion
Solving the quadratic equation 4x^2 – 5x – 12 = 0 using different methods yields a comprehensive understanding of quadratic equations. Each method has its own merits:
Factoring is easy if applicable and Completing the square offers insights into the structure of the equation. Also the quadratic formula is a reliable for all quadratic equations. Whereas Graphing provides a visual perspective.
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