Solving 42x2+-44x+94 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:
42x2+-44x+94=0.

There are no solutions in the real number domain.
There are two complex solutions: x = 0.52380952380952 + 1.4013275209107i, and x = 0.52380952380952 - 1.4013275209107i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           42x2+-44x+94=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)           \(x=--44\pm\frac{\sqrt{-44^2-4*42*94}}{2*42}\)

which simplifies to:
(4)           \(x=--44\pm\frac{\sqrt{1936-15792}}{84}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 1936 - 15792 = -13856.
(5)           \(x=--44\pm\frac{\sqrt{-13856}}{84}\)

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)           \(x=--44\pm\frac{117.7115117565i}{84}\)

This equation further simplifies to:
(7)           \(x=-\frac{--44}{84}\pm1.4013275209107i\)

Solving for x, we find two solutions which are both complex numbers:
x = 0.52380952380952 + 1.4013275209107i
  and
x = 0.52380952380952 - 1.4013275209107i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 42x2+-44x+94=0.






Notes

What is a quadratic equation? Any equation that can take the form:
ax2 + bx + c = 0,
where x is a variable of unknown value. A, b, and c are constants. A and b are referred to as coefficients. Further, it is worth pointing out that a cannot equal zero. 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.

Compared to solving a linear equation, solving a quadratic equation requires some more advanced mathematics.. Fortunately, any quadratic equation can readily be solved using the quadratic formula. The quadratic formula is:


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. Depending on the values of a, b, and c, both roots may be the same, producing one solution for x.

Why do we need to be able to solve quadratic equations? Quadratic equations are needed to calculate answers in many real-world fields, including physics, pharmacokinetics and business.

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.

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