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 can be written in the form: ax2 + bx + c = 0. In this equation, a, b, and c are constants. X is a variable which is not known. The constants a and b are called coefficients. Also, it is worth pointing out that a cannot equal zero in the equation ax2+bx+c=0. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

In contrast to solving a linear equation, solving a quadratic equation is a more complicated task. Fortunately, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be quickly 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. This is the quadratic formula:


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. Rarely, these two roots may equal each other, resulting in one solution for x.

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

In our equation, a cannot be zero. However, b can be zero, and so can c.

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