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 as: ax2 + bx + c = 0. In this equation, x is a variable which is not known. A, b, and c are constants. A and b are called coefficients. Interestingly, a cannot be equal to zero in the equation ax2+bx+c=0. If a equals 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.

In contrast to solving a linear equation, solving a quadratic equation is a more complicated task. 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:


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, the two roots may have the same value, resulting in one solution for x.

Why do we care about qudratic equations? Quadratic equations are needed to find answers in many real-world fields, including engineering, pharmacokinetics and architecture.

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.

We hope you find this quadratic equation solver 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 using Quadratic-Equation-Calculator.com.

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