Solving 3x2+54x+76 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:
3x2+54x+76=0.

There are two real solutions: x = -1.5389902381335, and x = -16.461009761866.

Here's how we found that solution:

You entered the following equation:
(1)           3x2+54x+76=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)           \(x=-54\pm\frac{\sqrt{54^2-4*3*76}}{2*3}\)

which simplifies to:
(4)           \(x=-54\pm\frac{\sqrt{2916-912}}{6}\)

Now, solving for x, we find two real solutions:
\(x=\frac{-54+44.766058571199}{6}\) = -1.5389902381335,
  and
\(x=\frac{-54-44.766058571199}{6}\) = -16.461009761866,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 3x2+54x+76=0.






Notes

An equation that can be written as bx+c=0 is called a linear equation. It has one unknown, x, and 2 constants, b and c. If this equation were also to include the square of x as an unknown, it would become a quadratic equation. What is a quadratic equation? A quadratic equation is any equation that has the form: ax2 + bx + c = 0, where a, b, and c are constants. X is an unknown. The constants a and b, are referred to as coefficients. It is worth noting that a cannot be equal to 0 in the equation ax2+bx+c=0. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Compared to solving a linear equation, solving a quadratic equation is a more complicated task. However, you have this handy-dandy quadratic equation calculator. Acutally, quadratic equations can be always solved using the quadratic formula, which is the same technique used by this quadratic equation calculator. Try it, and it will explain each of the steps to you. Here 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 equal, producing one solution for x.

Quadratic equations are an important part of mathematics. Quadratic equations are needed to compute answers to many real-world problems. The laws of motion is one example of an application of quadratic equations.

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, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for using Quadratic-Equation-Calculator.com.

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