# Solving 11x2+49x+15 using the Quadratic Formula

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

What is a quadratic equation? Any function ax2 + bx + c = 0. In this equation, a, b, and c are constants. X is an unknown. The constants a and b, are referred to as coefficients. Interestingly, a cannot be 0.

Solving a linear equation is pretty basic. Solving a quadratic equation requires some more advanced mathematics. Fortunately, any quadratic equation can always be solved using the quadratic formula. This is the quadratic formula:

Solving a quadratic equation will always result in 2 solutions for x. These solutions are called roots. These roots may both be real numbers or, they may both be complex numbers. Rarely, both roots may be the same, meaning there will only be one solution for x.

There are many uses for quadratic equations. Quadratic equations are needed to compute answers to many real-world problems. The contour of a parablolic dish antenna is one example of an application of quadratic equations.

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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