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

A free quadratic equation calculator that shows and explains each step in solving your quadratic equation.

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 function that can take the form:
ax2 + bx + c = 0.
\ In this equation, x is a variable which is not known. A, b, and c are constants. The constants a and b, are referred to as coefficients. Interestingly, a cannot be 0.

Calculating a solution to a quadratic equation may appear daunting, because both x and x2 are unknown. However, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be always solved using the quadratic formula, which is the same technique used by this quadratic equation solver. Try it, and it will explain each of the steps to you. The quadratic formula is:

Since there are always 2 solutions to a square root (one negative, one positive), solving the quadratic equation results in 2 values for x. The two solutions for x (which may be positive or negative, real or complex) are called roots. Rarely, both roots may be the same, producing one solution for x.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to compute answers to many real-world problems. The acceleration of an object as it falls to earth 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.

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