Solving 26x2+-74x+-46 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:
26x2+-74x+-46=0.

There are two real solutions: x = 3.3709934249386, and x = -0.52483957878471.

Here's how we found that solution:

You entered the following equation:
(1)           26x2+-74x+-46=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)           \(x=--74\pm\frac{\sqrt{-74^2-4*26*-46}}{2*26}\)

which simplifies to:
(4)           \(x=--74\pm\frac{\sqrt{5476--4784}}{52}\)

Now, solving for x, we find two real solutions:
\(x=\frac{--74+101.2916580968}{52}\) = 3.3709934249386,
  and
\(x=\frac{--74-101.2916580968}{52}\) = -0.52483957878471,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation 26x2+-74x+-46=0.






Notes

What is a quadratic equation? Any equation that can take the form:
ax2 + bx + c = 0,
where x is a variable of unknown value. A, b, and c are constants. A and b are called coefficients. Interestingly, a cannot be equal to 0 in the equation ax2+bx+c=0. If a is equal to 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 pretty straightforward. Solving a quadratic equation requires more work. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. Here 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. Under extraordinary circumstances, these two roots may have the same value, resulting in one solution for x.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to find answers to many real-world problems. The geometry of a parablolic dish antenna is one example of an application of quadratic equations.

Because equations can be rearranged without losing their meaning, sometimes you may see an equation that isn't written exactly this way, but it's still a quadratic equation. For example, you probably know that
ax2 + bx + c = 0 means exactly the same thing as 0 = c + bx + ax2. They're just written differently.
Here are some other examples of ways to write the quadratic equation. They all mean the same thing:
  (1)     \(ax^2+bx=d\), where d = -c
  (2)     \(x^2+bx-d=e\), where a=1 and d=e-c
  (3)     \(ax^2=ex+d\), where d=-c and e=-b
  (4)     \(\frac{x^2}{f}-d=ex\), where d=-c and e=-b and \(f=\frac{1}{a}\)
Look at each of the examples above. Do you understand why they are still quadratic equations, and how they can be rearranged to look like our familiar formula?


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