One number under the root sign tells you what kind of answer to expect.
Describe the roots
What kind of roots does 2x² − 4x + 5 = 0 have?
The discriminant is b² − 4ac = 16 − 4(2)(5) = 16 − 40 = −24. A negative discriminant means the square root of a negative number, which no real number gives, so there are no real roots. The parabola never touches the x-axis. You never had to solve the equation to know this.
D = b² − 4ac
| D | Roots | The graph |
|---|---|---|
| positive, a perfect square | two rational | crosses at two nice points |
| positive, not a perfect square | two irrational | crosses at two awkward points |
| zero | real and equal: one repeated root | touches the axis and turns |
| negative | no real roots (not real) | never reaches the axis |
With c negative, −4ac becomes positive and the discriminant grows. x² + 2x − 5 has D = 4 + 20 = 24, a completely different case from x² + 2x + 5.
Set the discriminant to zero and solve for the unknown.
4x² + kx + 9 = 0 has one root
find k
k² − 4(4)(9) = 0
D = 0
k² = 144
so k = ±12
Both values work: 4x² + 12x + 9 = (2x + 3)² and 4x² − 12x + 9 = (2x − 3)². Two answers is correct here, and giving only one is the usual slip.
sum = −ba · product = ca
3x² − 12x + 7 = 0
a = 3, b = −12, c = 7
sum = 123 = 4
−ba, and −(−12) = 12
product = 73
ca
This is a fast way to check roots you have already found. If your two answers do not add to −ba, one of them is wrong.
x² − (sum)x + (product) = 0
roots 5 and −2
write the equation
sum = 3, product = −10
add and multiply
x² − 3x − 10 = 0
note the minus before the sum
Check by factoring back: (x − 5)(x + 2) = x² − 3x − 10 ✓ The sum enters with a negative sign, which is where most errors come from.
The same three tools, used where the question is shaped differently.
The unknown is c: for what k does x² − 8x + k = 0 have equal roots?
(−8)² − 4(1)(k) = 0
D = 0
64 − 4k = 0
a straight line in k, not a square
k = 16
check: x² − 8x + 16 = (x − 4)² ✓
Only one answer this time: ± appears only when the unknown is the one being squared.
No x term: the sum and product of the roots of 2x² − 18 = 0
a = 2, b = 0, c = −18
the missing term means b = 0
sum = −02 = 0
−ba
product = −182 = −9
check: the roots are 3 and −3 ✓
With no x term the roots are opposites, so they always add to zero.
Fraction roots: write an equation with roots 12 and 3.
sum = 72, product = 32
add and multiply
x² − (72)x + 32 = 0
x² − (sum)x + product
2x² − 7x + 3 = 0
× 2 to clear the fractions
(2x − 1)(x − 3) = 0 ✓
factors back to the roots
Multiplying through by a whole number keeps the same roots and gives integer coefficients.
Expect "describe the roots" from the discriminant, the sum or product of the roots of an equation whose a is not 1, a value of k that gives equal roots, and an equation built from two given roots. The fastest route is to write a, b and c with signs and never solve the equation: D for the kind, −ba and ca for sum and product. The 10-second check: factor or expand your built equation back to its roots, or test that a k value makes D exactly zero.
Find the sum
What is the sum of the roots of 2x² + 9x − 5 = 0?
The sum is −ba = −92. A drops the minus sign, which the formula puts there deliberately. B is the product, ca = −52. Check by solving: the roots are 12 and −5, and 0.5 + (−5) = −4.5 = −92 ✓
Hint: sum is −ba and product is ca. The minus belongs to the sum, not the product.
