Solving a Quadratic Equation

Clyde wonders if there is a function in Excel that solves quadratic equations. The short answer is that there is no such worksheet function built into the program. You can, however, use a couple of formulas to solve a quadratic equation, or you can develop your own user-defined function.

This is the general form of a quadratic equation:

ax^2 + bx + c = 0

In the equation, values a, b, and c are constants and you need to solve for x. There are always two possible solutions for x, which means that there are two quadratic formulas that derive from the quadratic equation:

(-b+SQRT(b^2-4ac))/2a
(-b-SQRT(b^2-4ac))/2a

This is easy enough to place into a worksheet. Assuming that your values for a, b, and c are in cells A1, B1, and C1, you could use the following formulas:

=(-B1+SQRT(B1^2-4*A1*C1))/(2*A1)
=(-B1-SQRT(B1^2-4*A1*C1))/(2*A1)

You can, at this point, change the values in A1, B1, and C1 and end up with the answers you need. These formulas will work for any value where (B1^2)>(4*A1*C1). If this is not true, there are still answers, but you start getting into imaginary numbers. At this point things can get sticky in a hurry, and mere mortals run into the real possibility of their heads exploding. If you want to adjust your formulas to deal with imaginary numbers, you might find this short discussion helpful:

https://www.excelforum.com/excel-formulas-and-functions/1116652-how-to-view-complex-number-from-quadratic-equation.html

I should note that I've also seen people use Excel's Goal Seek tool to solve a quadratic equation, but for my money, the formulas described above work just fine. My feeling is that if you can use a formula to derive a solution, it seems to me that the whole need for doing the iterative work at which the Goal Seek tool is so adept is removed. If you prefer to work with the Goal Seek tool, however, you may find this web page helpful:

https://www.wikihow.com/Solve-a-Quadratic-Equation-Using-the-Goal-Seek-Feature-on-Microsoft-Excel

I mentioned earlier that you could build your own user-defined function to derive your solutions to a quadratic equation. Here is one that you could easily implement:

Function Quadratic(a As Double, b As Double, _
  c As Double, r As Integer) As Variant

    Quadratic = CVErr(xlErrValue)
    If r = 1 Then
        Quadratic = CVErr(xlErrNA)
        If b ^ 2 > (4 * a * c) Then
            Quadratic = (-b + Sqr(b ^ 2 - (4 * a * c))) / (2 * a)
        End If
    End If
    If r = 2 Then
        Quadratic = CVErr(xlErrNA)
        If b ^ 2 > (4 * a * c) Then
            Quadratic = (-b - Sqr(b ^ 2 - (4 * a * c))) / (2 * a)
        End If
    End If
End Function

The function requires four parameters: the values for A, B, and C, plus an indicator of whether you want the first (positive) answer or the second (negative) answer. Given three values in cells A1, B1, and C1, here's how you would get the second (negative) result in your worksheet:

=Quadratic(A1, B1, C1, 2)

If your indicator (the fourth parameter) is not 1 or 2, then the function returns a #VALUE! error. If the constants you provide for the first three parameters result in an imaginary answer, then the function returns a #NUM! error.

ExcelTips is your source for cost-effective Microsoft Excel training. This tip (13686) applies to Microsoft Excel 2007, 2010, 2013, 2016, 2019, and Excel in Microsoft 365.