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
Reset

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 an equation that takes the form:
ax2 + bx + c = 0,
where x is a variable which is not known, and a, b, and c are constants. A and b are called coefficients. Also, a cannot equal to zero in the equation ax2+bx+c=0. 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.

Solving a linear equation is rather basic. Solving a quadratic equation is not quite so simple. However, you have this handy-dandy quadratic equation calculator. Acutally, quadratic equations can be readily 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 have real-life applications. Quadratic equations are needed to calculate answers to many real-world problems. For example, to compute how an object will rise and fall due to Earth's gravity would require the use of s quadratic equation.

In our equation, a cannot be zero. However, b can be zero, and so can c.

We hope you find this quadratic equation solver useful. We encourage you to try it with different values, and to read the explanation for how to reach your answer. But, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for your interest in Quadratic-Equation-Calculator.com.

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