5 mistakes learners make with the discriminant, each one named and explained.
b squared minus 4ac decides how many real roots there are, and equal roots means it is zero.
The answer is correct in value but written as 2 root 8 rather than in its simplest form.
The value is right, so there is nothing to prompt a further step.
Question. k^2 = 64
A common answer. k = plus or minus 2 root 8
The answer. k = plus or minus 8
Why. 2 root 8 is 2 times 2 root 2, which is 4 root 2, and in this case the answer was a whole number anyway.
Always take the largest square factor out of a surd, and check whether it was a whole number all along.
x^2 + 6x becomes (x + 6)^2 rather than (x + 3)^2.
The 6 is the number in the expression and using it directly needs no extra thought.
Question. x^2 + 6x + 11
A common answer. (x + 6)^2 - 25
The answer. (x + 3)^2 + 2
Why. Expanding (x + 3)^2 gives x^2 + 6x + 9, which is where the 6x comes from.
Halve the x coefficient. Expanding your bracket back is the check that never fails.
The completed square ends with the wrong sign on the final number.
There are two subtractions in a row and one of them is easy to invert.
Question. x^2 + 6x + 11
A common answer. (x + 3)^2 - 2
The answer. (x + 3)^2 + 2
Why. (x + 3)^2 is x^2 + 6x + 9, and 11 is 2 more than 9, so 2 is added.
Expand your answer back. It has to reproduce the original constant exactly.
4ac is added instead of subtracted, or its sign is lost when a or c is negative.
Two minus signs in one expression, and one of them belongs to a coefficient.
Question. x^2 + 4x + 5 = 0
A common answer. 16 + 20 = 36
The answer. 16 - 20 = -4
Why. The formula subtracts 4ac, and here 4ac is positive 20.
Work out 4ac as its own number first, sign included, then subtract it.
The discriminant is worked out as b minus 4ac, with the square never applied.
The formula is remembered as b something minus 4ac, and the square is the small part.
Question. x^2 + 4x + 5 = 0
A common answer. 4 - 20 = -16
The answer. 16 - 20 = -4
Why. The discriminant is b squared minus 4ac, so the 4 is squared first.
Work out b squared as its own number and write it down before subtracting anything.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.