# 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, where a, b, and c are constants. X is a variable which is not known. The constants a and b, are referred to as coefficients. It should be noted that a cannot be zero. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Solving a linear equation is relatively basic. Solving a quadratic equation requires more work. However, 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: 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, the two roots may be the same, meaning there will only be one solution for x.

Quadratic equations are an important part of mathematics. Quadratic equations are needed to find answers to many real-world problems. For example, to compute how an object will rise and fall due to Earth's gravity would require the use of s quadratic equation.

Because equations can be rearranged without losing their meaning, sometimes you may see an equation that isn't written exactly this way, but it's still a quadratic equation. For example, you probably know that
ax2 + bx + c = 0 means exactly the same thing as 0 = c + bx + ax2. They're just written differently.
Here are some other examples of ways to write the quadratic equation. They all mean the same thing:
(1)     $$ax^2+bx=d$$, where d = -c
(2)     $$x^2+bx-d=e$$, where a=1 and d=e-c
(3)     $$ax^2=ex+d$$, where d=-c and e=-b
(4)     $$\frac{x^2}{f}-d=ex$$, where d=-c and e=-b and $$f=\frac{1}{a}$$
Look at each of the examples above. Do you understand why they are still quadratic equations, and how they can be rearranged to look like our familiar formula?

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