7 mistakes learners make with solving quadratics, each one named and explained.
A product is zero only when one of its factors is zero, so the roots are the values that make each bracket zero, with the sign flipped.
The formula gives x = -4 plus or minus a root, with the division by 2 applied to only part of it.
The 2a sits under a long expression and it is easy to divide only what is nearest.
Question. x^2 + 4x + 1 = 0
A common answer. x = -4 plus or minus root 3
The answer. x = -2 plus or minus root 3
Why. The whole numerator is divided by 2a, including the minus b.
Draw the fraction bar right across before you write anything above it.
The two roots come out symmetric about zero, because minus b never made it into the answer.
The root is the complicated part and it takes all the attention.
Question. x^2 - 3x - 2 = 0
A common answer. x = 1.56 or x = -1.56
The answer. x = 3.56 or x = -0.56
Why. The roots are spread around 1.5, not around 0, because b is not zero.
The midpoint of your two roots must equal -b/2a. That check catches this instantly.
x^2 + 6x + 11 becomes (x + 3)^2 + 11, with nothing taken off for the square.
The bracket is the part that changes, so the constant looks like it is just being carried along.
Question. x^2 + 6x + 11
A common answer. (x + 3)^2 + 11
The answer. (x + 3)^2 + 2
Why. (x + 3)^2 already contains a 9, so 9 has to come off the 11.
Expand your answer back every time. The constant is where this always shows up.
2x^2 + 5x - 3 = 0 is solved as though it were x^2 + 5x - 3, so the roots come out whole.
Every quadratic met first has a 1 in front, so the coefficient does not register as doing anything.
Question. 2x^2 + 5x - 3 = 0
A common answer. x = 1 or x = -3
The answer. x = 1/2 or x = -3
Why. The 2 splits one of the roots in half, which is why fractions appear.
Substitute your roots back in. It is the only check that catches this every time.
x^2 - 4x + 3 = 0 is given roots of 0 and 4, taken from the numbers on the page.
The numbers in the equation are the only ones available if the factorising has not happened.
Question. x^2 - 4x + 3 = 0
A common answer. x = 0 and x = 4
The answer. x = 1 and x = 3
Why. Substituting 4 gives 16 - 16 + 3, which is 3 and not 0.
Substitute every root you write down. It has to give zero.
A negative discriminant is written as positive, so a quadratic with no real roots appears to have two.
The subtraction is done and the minus sign is dropped when the answer is written down.
Question. x^2 + 4x + 5 = 0
A common answer. 16 - 20 = 4
The answer. 16 - 20 = -4
Why. 20 is larger than 16, so the answer has to be negative.
Check which of the two numbers is bigger before subtracting. That fixes the sign in advance.
x squared minus 9 is factorised as (x - 3)(x - 3).
Both patterns involve squares and one of them is far more famous, so it gets used for both.
Question. x^2 - 9
A common answer. (x - 3)(x - 3)
The answer. (x + 3)(x - 3)
Why. (x - 3)(x - 3) expands to x^2 - 6x + 9, which has a middle term and the wrong sign at the end.
A difference of two squares needs one plus and one minus, which is what makes the middle disappear.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.