Solving Quadratic Equation 4x ^ 2 – 5x – 12 = 0 with Formula

Solving a quadratic equation like (4x^2 – 5x – 12 = 0) involves finding values of (x) that make the equation true. Quadratic equations are of the form (ax^2 + bx + c = 0), where (a), (b), and (c) are constants. The solutions to these equations can be found using various methods, such as factoring, completing the square, or using the quadratic formula. In this article, we will use the quadratic formula and also explore an example through factoring.

Understanding the Quadratic Formula

The quadratic formula is given by:

[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} ]

This formula provides the solutions to any quadratic equation (ax^2 + bx + c = 0). The term (b^2 – 4ac) is known as the discriminant, and it determines the nature of the roots (real and distinct, real and equal, or complex).

Solving (4x^2 – 5x – 12 = 0) Using the Quadratic Formula

1. Identify coefficients: In our equation, (a = 4), (b = -5), and (c = -12).
2. Calculate the discriminant: (b^2 – 4ac = (-5)^2 – 4 \times 4 \times -12).
3. Substitute in the formula: Plug these values into the quadratic formula and solve for (x).
4. Find the two solutions: You’ll get two values of (x), which are the solutions to the equation.

Also Read:  Solve: x*x*x is equal to 2 x

Example

Let’s solve (4x^2 – 5x – 12 = 0) using the quadratic formula:

1. Calculate the discriminant:
   [ \Delta = (-5)^2 – 4 \times 4 \times -12 = 25 + 192 = 217 ]
2. Apply the formula:
   [ x = \frac{-(-5) \pm \sqrt{217}}{2 \times 4} = \frac{5 \pm \sqrt{217}}{8} ] This gives us two solutions:
   [ x_1 = \frac{5 + \sqrt{217}}{8} ]
   [ x_2 = \frac{5 – \sqrt{217}}{8} ]

Additional Example Through Factoring

Consider a simpler quadratic equation, (x^2 – 5x + 6 = 0).

1. Factor the equation: Find two numbers that multiply to (+6) and add up to (-5). These numbers are (-2) and (-3).
2. Set each factor equal to zero:
   [ x – 2 = 0 \Rightarrow x = 2 ]
   [ x – 3 = 0 \Rightarrow x = 3 ]

So, the solutions are (x = 2) and (x = 3).

Also Read: Solving the Equation: 5x – 12 = 0

Conclusion

The quadratic formula is a reliable method to find the roots of any quadratic equation. It’s especially useful when the equation is not easily factorable. Understanding this formula and how to apply it is crucial in algebra and prepares students for more advanced mathematical concepts. The provided examples illustrate how to apply this formula and also show the factoring method for simpler equations.