Solving 78x2+85x+-75 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:
78x2+85x+-75=0.

There are two real solutions: x = 0.57692307692308, and x = -1.6666666666667.

Here's how we found that solution:

You entered the following equation:
(1)           78x2+85x+-75=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)           \(x=-85\pm\frac{\sqrt{85^2-4*78*-75}}{2*78}\)

which simplifies to:
(4)           \(x=-85\pm\frac{\sqrt{7225--23400}}{156}\)

Now, solving for x, we find two real solutions:
\(x=\frac{-85+175}{156}\) = 0.57692307692308,
  and
\(x=\frac{-85-175}{156}\) = -1.6666666666667,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 78x2+85x+-75=0.






Notes

What is a quadratic equation? Any function that can be written as: ax2 + bx + c = 0, where x is an unknown, and a, b, and c are constants. A and b are referred to as coefficients. Also, a cannot be equal to zero 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 fairly simple. Solving a quadratic equation requires more work. Fortunately, any quadratic equation can reliably be solved using 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, both roots may be equal, producing one solution for x.

Quadratic equations are an important part of mathematics. Quadratic equations are needed to find answers to many real-world problems. For example, to calculate whether a braking car can stop fast enough to avoid hitting something would require the use of s quadratic equation.

The term "quadratic" comes from the Latin word quadratum, which means "square." Why? Because what defines a quadratic equation is the inclusion of some variable squared. In our equation above, the term x2 (x squared) is what makes this equation quadratic.

We this quadratic equation calculator is useful to you. We encourage you to plug in different values for a, b, and c. But, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for your interest in Quadratic-Equation-Calculator.com.

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