The general form of a quadratic equation is ax^2 + bx + c = 0, where "a," "b," and "c" are coefficients. In our case, the quadratic equation is 4x^2 – 5x – 12 = 0. Here, "a" is 4, "b" is -5, and "c" is -12.

The Discriminant

To determine the nature of the solutions, we calculate the discriminant, given by the formula Δ = b^2 - 4ac. The discriminant provides insights into the number and type of solutions the quadratic equation possesses.

For our equation, Δ = (-5)^2 - 4(4)(-12) = 25 + 192 = 217. Since Δ is positive, the quadratic equation has two distinct real solutions.

Solving the Quadratic Equation

�=−�±Δ2�x=2a−b±Δ​​

Substituting the values from our equation, we get:

�=5±2178x=85±217​​

Therefore, the two solutions are:

�1=5+2178x1​=85+217​​

�2=5−2178x2​=85−217​​

These solutions may appear complex due to the presence of the square root of 217, but they accurately represent the values of "x" that satisfy the given quadratic equation.

Real-World Applications

Quadratic equations are not merely abstract mathematical concepts; they find applications in various real-world scenarios. From physics to engineering, economics to biology, quadratic equations help model and solve problems involving relationships between variables.

Understanding how to solve quadratic equations is a valuable skill that empowers individuals to analyze and solve problems in diverse fields.

In conclusion, the quadratic equation 4x^2 – 5x – 12 = 0 can be solved using the quadratic formula, providing two distinct real solutions. The discriminant plays a crucial role in determining the nature of these solutions. Recognizing the applications of quadratic equations in real-world scenarios enhances the appreciation for their significance in mathematics and beyond.