Solving 30x2+96x+-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:
30x2+96x+-75=0.

There are two real solutions: x = 0.6494443758404, and x = -3.8494443758404.

Here's how we found that solution:

You entered the following equation:
(1)           30x2+96x+-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)           \(x=-96\pm\frac{\sqrt{96^2-4*30*-75}}{2*30}\)

which simplifies to:
(4)           \(x=-96\pm\frac{\sqrt{9216--9000}}{60}\)

Now, solving for x, we find two real solutions:
\(x=\frac{-96+134.96666255042}{60}\) = 0.6494443758404,
  and
\(x=\frac{-96-134.96666255042}{60}\) = -3.8494443758404,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 30x2+96x+-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 an unknown. The constants a and b, are referred to as coefficients. Furthermore, it is worth pointing out that a cannot be 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.

Solving a linear equation is simple. Solving a quadratic equation is less so. However, any quadratic equation can reliably be solved using the quadratic formula. The quadratic formula is:


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

Why do we care about qudratic equations? Quadratic equations are needed to compute answers to many real-world problems. For example, to calculate the path of an accelerating object 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 hope you find this quadratic equation calculator useful. 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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