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

A quadratic equation is any function that has the form: ax2 + bx + c = 0, where x is a variable which is not known, and a, b, and c are constants. A and b are called coefficients. Also, it should be mentioned that a cannot equal to 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 relatively basic. 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 written:


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 be equal, meaning there will only be one solution for x.

Quadratic equations are more than just mathematical chores we have to endure. Quadratic equations are needed to find answers to many real-world problems. For example, to calculate how an object will rise and fall due to Earth's gravity would require the use of s quadratic equation.

As mentioned above, in the equation ax2+bx+c=0, a cannot be zero. If a were 0, then ax2 = 0x2 = 0 for any value of x, so our equation becomes 0 + bx + c = 0, which is the same as bx + c = 0, which is no longer a quadratic equation. In fact, bx + c = 0 is a linear equation, which is much simpler to solve than a quadratic equation.

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