Solving Quadratic Equations: -4x^2-5x-12=0

Polynomial conditions of degree two, and quadratic conditions are now and again seen in numerous areas of math and material science. Finding the values of the variable that fulfill quadratic equations is the process of solving them. Here the equation is -4x^2-5x-12=0. We’ll work through the solutions one by one.

Step 1: Identify the Coefficients

Find the quadratic equation’s coefficients before we begin solving it. The coefficients in our equation, -4x^2-5x-12=0, are:

Coefficient of x^2 is -4.
Coefficient x is -5.
It is a continuous term -12.

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Step 2: Use the Quadratic Formula

Quadratic issues can be capably settled with the utilization of the quadratic recipe. It ensures that the recipe under can be used to find the solutions for x in any quadratic state of the design ax^2+bx+c=0:

x = (-b ± √(b^2-4ac)) / (2a)

Let us now enter our equation’s coefficients into the quadratic formula:

x = (-(-5) ± √((-5)^2-4(-4)(-12))) / (2(-4))

Simplifying further:

x = (5 ± √(25-192)) / (-8)

x = (5 ± √(-167)) / (-8)

Step 3: Determine the Nature of the Solutions

We run across a difficulty at this moment. Within the square root, -167, is a negative word. Complex answers arise when one takes the square root of a negative quantity. In this case, we are working with real numbers, hence the quadratic equation -4x^2-5x-12=0 has no actual solutions.

However, if you are interested in finding complex solutions, we can proceed with the calculations. Let’s continue.

Step 4: Simplify the Square Root of a Negative Number

To work on the square foundation of a negative number, we can present the fanciful unit, signified by I. The fanciful unit I is characterized as the square base of – 1.

Let’s rewrite the equation as:

x = (5 ± √(167)i) / (-8)

Now we have two complex solutions for x:

x = (5 + √(167)i) / (-8)

x = (5 – √(167)i) / (-8)

These solutions involve the imaginary unit i and represent complex numbers.

Conclusion

The quadratic equation -4x^2-5x-12=0 has no actual solutions, to sum up. If you would want more complicated answers, the quadratic formula can be used to solve the equation. The answers illustrate complex numbers and use the imaginary unit i.

Remember that the discriminant—the term within the square root—determines the kind of solutions for quadratic equations. For one to comprehend the behavior of the equation, one must examine the nature of the solutions.