Solving 72x2+72x+-73 using the Quadratic Formula

A free quadratic equation calculator that shows and explains each step in solving your quadratic equation.


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:
72x2+72x+-73=0.

There are two real solutions: x = 0.62422813026934, and x = -1.6242281302693.

Here's how we found that solution:

You entered the following equation:
(1)           72x2+72x+-73=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)           \(x=-72\pm\frac{\sqrt{72^2-4*72*-73}}{2*72}\)

which simplifies to:
(4)           \(x=-72\pm\frac{\sqrt{5184--21024}}{144}\)

Now, solving for x, we find two real solutions:
\(x=\frac{-72+161.88885075878}{144}\) = 0.62422813026934,
  and
\(x=\frac{-72-161.88885075878}{144}\) = -1.6242281302693,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 72x2+72x+-73=0.






Notes

An equation that can be written as bx+c=0 is called a linear equation. It has one unknown, x, and 2 constants, b and c. If this equation were also to include the square of x as an unknown, it would become a quadratic equation. A quadratic equation is an equation that can be written in the form: ax2 + bx + c = 0, where x is unknown. A, b, and c are constants. A and b are called coefficients. Further, it is worth mentioning that a cannot equal to zero. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Calculating a solution to a quadratic equation may seem challenging. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is 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. Rarely, the two roots may be the same, meaning there will only be one solution for x.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to calculate answers in many real-world fields, including engineering, 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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