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

What is a quadratic equation? Any equation that can take the form: ax2 + bx + c = 0, where x is an unknown, and a, b, and c are constants. A and b are called coefficients. Furthermore, it should be pointed out that a cannot be equal to 0 in the equation ax2+bx+c=0.

Solving a linear equation is pretty basic. Solving a quadratic equation is not quite so straightforward. Fortunately, you have this handy-dandy quadratic equation solver. All kidding aside, quadratic equations can be quickly 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. 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. Depending on the values of a, b, and c, both roots may equal each other, producing one solution for x.

Quadratic equations are an important part of mathematics. Quadratic equations are needed to compute answers in many real-world fields, including physics, biology and architecture.

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?


We hope you find this quadratic equation solver useful. We hope the explanations showing how you can solve the equation yourself are educational and helpful. 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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