# 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 any function that can be written in the form:
ax2 + bx + c = 0,
where x is an unknown. A, b, and c are constants. A and b are called coefficients. Interestingly, a cannot equal zero in the equation ax2+bx+c=0.

Solving a linear equation is pretty simple. Solving a quadratic equation is less so. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. The quadratic formula is written:

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 equal each other, resulting in one solution for x.

Why do we need to be able to solve quadratic equations? Quadratic equations are needed to compute answers to many real-world problems. For example, to calculate the path of an accelerating object would require the use of s quadratic equation.

Because equations can be rearranged without losing their meaning, sometimes you may see an equation that isn't written exactly this way, but it's still a quadratic equation. For example, you probably know that
ax2 + bx + c = 0 means exactly the same thing as 0 = c + bx + ax2. They're just written differently.
Here are some other examples of ways to write the quadratic equation. They all mean the same thing:
(1)     $$ax^2+bx=d$$, where d = -c
(2)     $$x^2+bx-d=e$$, where a=1 and d=e-c
(3)     $$ax^2=ex+d$$, where d=-c and e=-b
(4)     $$\frac{x^2}{f}-d=ex$$, where d=-c and e=-b and $$f=\frac{1}{a}$$
Look at each of the examples above. Do you understand why they are still quadratic equations, and how they can be rearranged to look like our familiar formula?

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