Solving 11x2+-23x+-13 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+-23x+-13=0.

There are two real solutions: x = 2.5536963657306, and x = -0.46278727482155.

Here's how we found that solution:

You entered the following equation:
(1)           11x2+-23x+-13=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)           \(x=--23\pm\frac{\sqrt{-23^2-4*11*-13}}{2*11}\)

which simplifies to:
(4)           \(x=--23\pm\frac{\sqrt{529--572}}{22}\)

Now, solving for x, we find two real solutions:
\(x=\frac{--23+33.181320046074}{22}\) = 2.5536963657306,
  and
\(x=\frac{--23-33.181320046074}{22}\) = -0.46278727482155,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 11x2+-23x+-13=0.






Notes

What is a quadratic equation? A quadratic equation is an function that takes the form: ax2 + bx + c = 0, where x is an unknown. A, b, and c are constants. The constants a and b, are referred to as coefficients. Interestingly, a cannot be 0 in the equation ax2+bx+c=0. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Solving a linear equation is simple. Solving a quadratic equation is not as straightforward. 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:


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. Under extraordinary circumstances, the two roots may have the same value, producing one solution for x.

Quadratic equations are more than just mathematical mumbo-jumbo Quadratic equations are needed to compute answers to many real-world problems. For example, to compute whether a braking car can stop fast enough to avoid hitting something would require the use of s quadratic equation.

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

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