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 function that can be written in the form:
ax2 + bx + c = 0,
where x is a variable of unknown value. A, b, and c are constants. A and b are referred to as coefficients. It is worth pointing out that a cannot be zero. 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 quadratic equation may appear daunting, because both x and x2 are unknown. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. The quadratic formula is:


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, these two roots may equal each other, resulting in one solution for x.

Quadratic equations are important. Quadratic equations are needed to find answers in many real-world fields, including physics, biology 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.

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