# Solving 65x2+61x+-91 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:
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

What is a quadratic equation? A quadratic equation is an equation
ax2 + bx + c = 0,
where a, b, and c are constants. X is an unknown. The constants a and b, are referred to as coefficients. Interestingly, a cannot equal zero 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.

Calculating a solution to a quadratic equation may appear daunting, because both x and x2 are unknown. However, you have this handy-dandy quadratic equation calculator. Acutally, quadratic equations can be always 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. Here 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. Rarely, the two roots may be equal, resulting in one solution for x.

So what? Why do we care about qudratic equations? Quadratic equations are needed to find 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.

The term "quadratic" comes from the Latin word quadratum, which means "square." Why? Because what defines a quadratic equation is the inclusion of some variable squared. In our equation above, the term x2 (x squared) is what makes this equation quadratic.

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