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:

= 0

You entered:

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:

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,
\(x=\frac{--66-157.11142542794}{132}\) = -0.69023807142381,

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


What is a quadratic equation? A quadratic equation is an equation ax2 + bx + c = 0. In this equation, x is a variable of unknown value. A, b, and c are constants. The constants a and b, are referred to as coefficients. It should be mentioned that a cannot be zero. If a=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 quadratic equation can seem challenging. However, 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:

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. Under extraordinary circumstances, these two roots may be equal, resulting in one solution for x.

Why do we care about qudratic equations? Quadratic equations are needed to find answers in many real-world fields, including physics, biology and business.

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.

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