Solving 66x2+-66x+-77 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:
66x2+-66x+-77=0.

There are two real solutions: x = 1.6902380714238, and x = -0.69023807142381.

Here's how we found that solution:

You entered the following equation:
(1)           66x2+-66x+-77=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)           \(x=--66\pm\frac{\sqrt{-66^2-4*66*-77}}{2*66}\)

which simplifies to:
(4)           \(x=--66\pm\frac{\sqrt{4356--20328}}{132}\)

Now, solving for x, we find two real solutions:
\(x=\frac{--66+157.11142542794}{132}\) = 1.6902380714238,
  and
\(x=\frac{--66-157.11142542794}{132}\) = -0.69023807142381,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 66x2+-66x+-77=0.






Notes

What is a quadratic equation? 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, and a, b, and c are constants. A and b are referred to as coefficients. Interestingly, a cannot equal to zero. If a is 0, then ax2=0, and the equation becomes 0+bx+c=0, or bx+c=0. The equation bx+c=0 is a linear equation, and not a quadratic equation.

Calculating a solution to a quadratic equation may appear daunting, because both x and x2 are unknown. Fortunately, any quadratic equation can quickly be solved using the quadratic formula. This 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, both roots may be the same, meaning there will only be one solution for x.

Quadratic equations are more than just mathematical flights of fantasy Quadratic equations are needed to find answers to many real-world problems. The geometry of a parablolic dish antenna 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.

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