# Solving 65x2+61x+-91 using the Quadratic Formula

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:
65x2+61x+-91=0.

There are two real solutions: x = 0.80363116939137, and x = -1.7420927078529.

## Here's how we found that solution:

You entered the following equation:
(1)           65x2+61x+-91=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)           $$x=-61\pm\frac{\sqrt{61^2-4*65*-91}}{2*65}$$

which simplifies to:
(4)           $$x=-61\pm\frac{\sqrt{3721--23660}}{130}$$

Now, solving for x, we find two real solutions:
$$x=\frac{-61+165.47205202088}{130}$$ = 0.80363116939137,
and
$$x=\frac{-61-165.47205202088}{130}$$ = -1.7420927078529,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 65x2+61x+-91=0.

### Notes

A quadratic equation is an equation that can be written as:
ax2 + bx + c = 0,
where a, b, and c are constants. X is unknown. The constants a and b, are referred to as coefficients. Furthermore, it should be mentioned that a cannot be 0 in the equation ax2+bx+c=0. 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.

In contrast to solving a linear equation, solving a quadratic equation requires some more advanced mathematics.. Fortunately, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be always solved using the quadratic formula, which is the same technique used by this quadratic equation solver. Try it, and it will explain each of the steps to you. This is the quadratic formula:

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. Depending on the values of a, b, and c, the two roots may be equal, meaning there will only be one solution for x.

There are many uses for quadratic equations. Quadratic equations are needed to calculate answers to many real-world problems. The geometry of a parablolic mirror 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 calculator 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.