Solving 11x2+49x+15 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:
11x2+49x+15=0.

There are two real solutions: x = -0.33066854136136, and x = -4.1238769131841.

Here's how we found that solution:

You entered the following equation:
(1)           11x2+49x+15=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)           \(x=-49\pm\frac{\sqrt{49^2-4*11*15}}{2*11}\)

which simplifies to:
(4)           \(x=-49\pm\frac{\sqrt{2401-660}}{22}\)

Now, solving for x, we find two real solutions:
\(x=\frac{-49+41.72529209005}{22}\) = -0.33066854136136,
  and
\(x=\frac{-49-41.72529209005}{22}\) = -4.1238769131841,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 11x2+49x+15=0.






Notes

A quadratic equation is any equation that takes the form: ax2 + bx + c = 0, where x is a variable which is not known. A, b, and c are constants. A and b are called coefficients. Further, it is worth mentioning that 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 linear equation is basic. Solving a quadratic equation requires some more advanced mathematics. Fortunately, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be always 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. Here 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. Depending on the values of a, b, and c, these two roots may equal each other, 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 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?


We hope you find this quadratic equation calculator useful. We hope the explanations showing how you can solve the equation yourself are educational and helpful. But, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for using Quadratic-Equation-Calculator.com.

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