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
Reset

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

A quadratic equation is any function that takes the form:
ax2 + bx + c = 0.
\ In this equation, x is a variable which is not known. A, b, and c are constants. A and b are called coefficients. It should be noted that a cannot equal to zero in the equation ax2+bx+c=0.

Solving a linear equation is simple. Solving a quadratic equation is less simple. However, you have this handy-dandy quadratic equation calculator. All kidding aside, quadratic equations can be reliably 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. The quadratic formula is:


When you compute a solution to a quadratic equation, you will always find 2 values for x, called "roots". These roots may both be real numbers or, they may both be complex numbers. Rarely, both roots may have the same value, producing one solution for x.

Quadratic equations are more than just mathematical chores we have to endure. Quadratic equations are needed to find 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 term "quadratic" comes from the Latin word quadratum, which means "square." Why? Because what defines a quadratic equation is the inclusion of some variable squared. In our equation above, the term x2 (x squared) is what makes this equation quadratic.

We this quadratic equation calculator is useful to you. 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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