# Solving 12x2+72x+54 using the Quadratic Formula

For your equation of the form "ax2 + bx + c = 0," enter the values for a, b, and c:

 a x2 + b x + c = 0
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You entered:
12x2+72x+54=0.

There are two real solutions: x = -0.87867965644036, and x = -5.1213203435596.

## Here's how we found that solution:

You entered the following equation:
(1)           12x2+72x+54=0.

For any quadratic equation ax2 + bx + c = 0, one can solve for x using the following equation, which is known as the quadratic formula:
(2)

In the form above, you specified values for the variables a, b, and c. Plugging those values into Eqn. 1, we get:
(3)           $$x=-72\pm\frac{\sqrt{72^2-4*12*54}}{2*12}$$

which simplifies to:
(4)           $$x=-72\pm\frac{\sqrt{5184-2592}}{24}$$

Now, solving for x, we find two real solutions:
$$x=\frac{-72+50.911688245431}{24}$$ = -0.87867965644036,
and
$$x=\frac{-72-50.911688245431}{24}$$ = -5.1213203435596,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 12x2+72x+54=0.

### Notes

A quadratic equation is an equation that can be written as:
ax2 + bx + c = 0,
where x is a variable which is not known, and a, b, and c are constants. The constants a and b are called coefficients. Additionally, it is worth mentioning that a cannot equal to zero in the equation ax2+bx+c=0. If a=0, then ax2=0, and the equation becomes 0+bx+c=0, or bx+c=0. The equation bx+c=0 is a linear equation, and not a quadratic equation.

Solving a linear equation is basic. Solving a quadratic equation is less simple. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. This is the quadratic formula:

Solving a quadratic equation will always result in 2 solutions for x. These solutions are called roots. These roots may both be real numbers or, they may both be complex numbers. Rarely, these two roots may be the same, producing one solution for x.

Quadratic equations are important. Quadratic equations are needed to find answers to many real-world problems. The contour of a parablolic dish antenna is one example of an application of quadratic equations.

The quadratic equation calculator on this website uses the quadratic formula to solve your quadratic equations, and this is a reliable and relatively simple way to do it. But there are other ways to solve a quadratic equation, such as completing the square or factoring.

We this quadratic equation solver is useful to you. We encourage you to try it with different values, and to read the explanation for how to reach your answer. But we totally understand if you just want to use it to find the answers you're looking for. Thank you for using Quadratic-Equation-Calculator.com.