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

A quadratic equation is any equation
ax2 + bx + c = 0.
\ In this equation, a, b, and c are constants. X is an unknown. A and b are referred to as coefficients. It should be pointed out that a cannot equal to zero.

Finding a solution to a quadratic equation may appear daunting, because both x and x2 are unknown. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. 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. Under extraordinary circumstances, these two roots may be the same, producing one solution for x.

Why do we need to be able to solve quadratic equations? 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 quadratic formula has been known for centuries. Brahmagupta, a mathematician from India, first described the quadratic formula as a means to calculate solutions to quadratic equations in the 7th Century AD.

We hope you find this quadratic equation solver useful. We encourage you to plug in different values for a, b, and c. 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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