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

A quadratic equation is an equation that can be written as:
ax2 + bx + c = 0.
\ In this equation, a, b, and c are constants. X is unknown. A and b are referred to as coefficients. Interestingly, a cannot be equal to 0. If a is 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 requires some more advanced mathematics.. However, there are a number of methods for solving quadratic equations. One of the most widely used is 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. Depending on the values of a, b, and c, the two roots may be the same, meaning there will only be one solution for x.

There are many uses for quadratic equations. Quadratic equations are needed to find answers to many real-world problems. The distance before a vehicle can stop once you hit the brakes is one example of an application of quadratic equations.

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 this quadratic equation calculator is useful to you. 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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