# Solving 95x2+-59x+8 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:
95x2+-59x+8=0.

There are two real solutions: x = 0.42105263157895, and x = 0.2.

## Here's how we found that solution:

You entered the following equation:
(1)           95x2+-59x+8=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)           $$x=--59\pm\frac{\sqrt{-59^2-4*95*8}}{2*95}$$

which simplifies to:
(4)           $$x=--59\pm\frac{\sqrt{3481-3040}}{190}$$

Now, solving for x, we find two real solutions:
$$x=\frac{--59+21}{190}$$ = 0.42105263157895,
and
$$x=\frac{--59-21}{190}$$ = 0.2,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 95x2+-59x+8=0.

### Notes

A quadratic equation is an equation that takes the form:
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 equal 0 in the equation ax2+bx+c=0.

Calculating a solution to a quadratic equation may appear daunting, because both x and x2 are unknown. However, any quadratic equation can quickly be solved using the quadratic formula. The quadratic formula is written:

Since there are always 2 solutions to a square root (one negative, one positive), solving the quadratic equation results in 2 values for x. The two solutions for x (which may be positive or negative, real or complex) are called roots. Depending on the values of a, b, and c, these two roots may be equal, resulting in one solution for x.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to compute answers to many real-world problems. For example, to compute 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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