Solving 65x2+61x+-91 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:
65x2+61x+-91=0.

There are two real solutions: x = 0.80363116939137, and x = -1.7420927078529.

Here's how we found that solution:

You entered the following equation:
(1)           65x2+61x+-91=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)           \(x=-61\pm\frac{\sqrt{61^2-4*65*-91}}{2*65}\)

which simplifies to:
(4)           \(x=-61\pm\frac{\sqrt{3721--23660}}{130}\)

Now, solving for x, we find two real solutions:
\(x=\frac{-61+165.47205202088}{130}\) = 0.80363116939137,
  and
\(x=\frac{-61-165.47205202088}{130}\) = -1.7420927078529,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 65x2+61x+-91=0.






Notes

A quadratic equation is an function that can be written in the form:
ax2 + bx + c = 0,
where a, b, and c are constants. X is a variable which is not known. A and b are referred to as coefficients. Further, it should be pointed out that a cannot equal to zero. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Solving a linear equation is pretty basic. Solving a quadratic equation requires more work. However, there are a number of methods for solving quadratic equations. One of the most widely used is 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. Under extraordinary circumstances, the two roots may have the same value, producing one solution for x.

Why do we need to be able to solve quadratic equations? Quadratic equations are needed to compute answers in many real-world fields, including physics, biology and architecture.

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

We hope you find this quadratic equation calculator useful. We encourage you to plug in different values for a, b, and c. But we totally understand if you just want to use it to find the answers you're looking for. Thank you for your interest in Quadratic-Equation-Calculator.com.

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