The equation in question is a quadratic equation, which means it is a polynomial equation of the second degree. Quadratic equations are characterized by having a variable raised to the power of 2 (x^2) as the highest power.
The equation 4x^2 – 5x – 12 = 2 is a specific type of quadratic equation known as a standard form quadratic equation, which can be written as ax^2 + bx + c = 0. In our case, ‘a’ is 4, ‘b’ is -5, and ‘c’ is -12 – 2, which simplifies to -14.
Applying the Quadratic Formula
Now that we’ve identified our values for ‘a,’ ‘b,’ and ‘c,’ we can proceed to solve the equation using the quadratic formula. The quadratic formula is a powerful tool for finding the solutions to any quadratic equation of the form ax^2 + bx + c = 0. It is expressed as:
x = (-b ± √(b^2 – 4ac)) / (2a)
In our case, with ‘a’ as 4, ‘b’ as -5, and ‘c’ as -14, we can plug these values into the formula:
x = (-(-5) ± √((-5)^2 – 4 * 4 * (-14))) / (2 * 4)
Simplifying this further:
x = (5 ± √(25 + 224)) / 8
x = (5 ± √249) / 8
Now, we have two possible solutions for ‘x’ obtained by the positive and negative roots of the equation. These solutions are:
x₁ = (5 + √249) / 8
x₂ = (5 – √249) / 8
Calculating the Solutions
To find the exact numerical values of x₁ and x₂, we can perform the calculations:
x₁ ≈ 2.6119
x₂ ≈ -1.1369
So, the solutions to the equation 4x^2 – 5x – 12 = 2 are approximately x₁ ≈ 2.6119 and x₂ ≈ -1.1369.
Interpretation and Application
Now that we have successfully solved the equation, let’s discuss its significance and potential applications. Quadratic equations, such as the one we just solved, are ubiquitous in fields like physics, engineering, economics, and computer science. They are used to model various real-world phenomena, including projectile motion, optimization problems, and financial analysis.
In the case of our equation, understanding its solutions can help us determine the values of ‘x’ that satisfy the equation, which may have practical implications in specific scenarios. For example, in a physics problem involving the motion of a projectile, ‘x’ could represent time, and the solutions would provide the times at which the projectile reaches a certain height.
