Quadratic Equations Solver

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Valid Inputs

Valid integer coefficients for  \(ax^2+bx+c\):  

\(a\):  \(\{1-9\}\),   \(b\):  \(\{0-99\}\),   \(c\):  \(\{0-99\}\)

The signs can be toggled between \(+\) and \(-\) by clicking the one which you want to change.
As stated in the applet, the \(x^2\) coefficient will be changed to 1 if completing the square or factorising are used. This is to save on space in the workings area as there wasn't room to use extra lines for fraction simplification. Unless you can divide through by the \(x^2\) coefficient to leave integers in the other two terms (in which case you can do that and THEN use this applet), it's generally easier to use the quadratic formula anyway!

The Random Generator

This button will randomly choose signs and coefficients in the valid ranges stated above.

What if I want to input a negative \(x^2\) coefficient?

To keep the applet more simple, you cannot do this. However, if you simply multiply through each term in your question by \((-1)\) then you can solve that way, for example: \[-x^2 + 4x - 3 = 0 \Longrightarrow x^2 - 4x + 3 = 0\]

Getting an Output

Click one of the three green buttons below the input section, depending which type of workings you want displayed. For revision, explainations and examples on each method, check out their pages here:

The Quadratic Formula & Completing the Square (CTS)

Both of these methods have similar outputs; the worked solution and answer. Unlike the factorising which gives the final line "\(x = \) __ or __", these methods give the two roots of the equation in the form "\(x = a \pm b \)". If there is no "\(\pm\)" to suggest two distinct roots then it should state that it is a single repeated root.

It should be noted that for CTS, fractions displayed as \(a/b\) should be seen the same as fractions displayed \(\frac{a}{b}\), it is merely because of space constraints that some simple fractions are displayed inline like the prior example.


The factorising method will only give solutions if the roots are integers. If you see the error message:

"None, this won't factorise 'nicely'
so maybe try another method"

Then do as it suggests! The error message could be for one of two reasons; either there are no roots (try checking the discriminant) or the roots have a surd or fraction part.

HELP?! "There are no real roots to this equation"

This message will have appeared because at some point in the workings there was a square root of a negative number. This means the roots will be complex (a further maths topic), so we say there are no real roots. This could have been found out before we did any workings using the discriminant.


If you find any issues, let me know on Twitter: @nextlevelmaths.