Solving 36x2+91x+75 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:
36x2+91x+75=0.

There are no solutions in the real number domain.
There are two complex solutions: x = -1.2638888888889 + 0.6970783384072i, and x = -1.2638888888889 - 0.6970783384072i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           36x2+91x+75=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)           \(x=-91\pm\frac{\sqrt{91^2-4*36*75}}{2*36}\)

which simplifies to:
(4)           \(x=-91\pm\frac{\sqrt{8281-10800}}{72}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 8281 - 10800 = -2519.
(5)           \(x=-91\pm\frac{\sqrt{-2519}}{72}\)

This means that our solution will require finding the square root of a negative number. There is no real number solution for this, so our solution will be a complex number (that is, it will involve the imaginary number i, defined as the square root of -1.).
Let's calculate the square root:
(6)           \(x=-91\pm\frac{50.189640365318i}{72}\)

This equation further simplifies to:
(7)           \(x=-\frac{-91}{72}\pm0.6970783384072i\)

Solving for x, we find two solutions which are both complex numbers:
x = -1.2638888888889 + 0.6970783384072i
  and
x = -1.2638888888889 - 0.6970783384072i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 36x2+91x+75=0.






Notes

What is a quadratic equation? Any equation that takes the form:
ax2 + bx + c = 0,
where x is unknown, and a, b, and c are constants. A and b are called coefficients. Interestingly, a cannot equal zero in the equation ax2+bx+c=0. If a is equal to 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.

Solving a quadratic equation may appear daunting, because both x and x2 are unknown. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. Here 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. Under extraordinary circumstances, these two roots may equal each other, meaning there will only be one solution for x.

Quadratic equations are important. Quadratic equations are needed to calculate answers in many real-world fields, including physics, biology and business.

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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