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

What is a quadratic equation? Any equation
ax2 + bx + c = 0.
\ In this equation, a, b, and c are constants. X is a variable which is not known. The constants a and b are called coefficients. Further, it is worth pointing out that a cannot be equal to 0 in the equation ax2+bx+c=0.

Solving a linear equation is straightforward. Solving a quadratic equation requires some more advanced mathematics. Fortunately, you have this handy-dandy quadratic equation solver. All kidding aside, quadratic equations can be readily 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. 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, the two roots may have the same value, meaning there will only be one solution for x.

Why do we need to be able to solve quadratic equations? Quadratic equations are needed to calculate 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.

As mentioned above, in the equation ax2+bx+c=0, a cannot be zero. If a were 0, then ax2 = 0x2 = 0 for any value of x, so our equation becomes 0 + bx + c = 0, which is the same as bx + c = 0, which is no longer a quadratic equation. In fact, bx + c = 0 is a linear equation, which is much simpler to solve than a quadratic equation.

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 your interest in Quadratic-Equation-Calculator.com.

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