# Solving 65x2+-13x+-63 using the Quadratic Formula

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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+-13x+-63=0.

There are two real solutions: x = 1.0895608971816, and x = -0.88956089718156.

## Here's how we found that solution:

You entered the following equation:
(1)           65x2+-13x+-63=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)           $$x=--13\pm\frac{\sqrt{-13^2-4*65*-63}}{2*65}$$

which simplifies to:
(4)           $$x=--13\pm\frac{\sqrt{169--16380}}{130}$$

Now, solving for x, we find two real solutions:
$$x=\frac{--13+128.6429166336}{130}$$ = 1.0895608971816,
and
$$x=\frac{--13-128.6429166336}{130}$$ = -0.88956089718156,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 65x2+-13x+-63=0.

### Notes

A quadratic equation is an function that can be written as:
ax2 + bx + c = 0.
\ In this equation, x is a variable which is not known. A, b, and c are constants. The constants a and b are called coefficients. Also, it should be mentioned that a cannot be equal to 0.

Compared to solving a linear equation, solving a quadratic equation requires a few more steps. However, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be reliably 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. The quadratic formula is: 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. Depending on the values of a, b, and c, these two roots may have the same value, meaning there will only be one solution for x.

Quadratic equations are more than just mathematical chores we have to endure. Quadratic equations are needed to find answers to many real-world problems. The contour of a parablolic dish antenna 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 using Quadratic-Equation-Calculator.com.

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