One formula solves every quadratic. Writing a, b and c down first is what makes it reliable.
Identify the coefficients
Written in standard form with a positive leading coefficient, what are a, b and c for 5x − 3 = 2x²?
Move everything to the side that keeps 2x² positive: 2x² − 5x + 3 = 0, so a = 2, b = −5, c = 3. B keeps the signs from the original arrangement without rearranging. D forgets that moving −3 across makes it +3. Every sign must come from the same, final arrangement.
ax² + bx + c = 0, with every sign part of its coefficient.
| Equation | Standard form | a, b, c |
|---|---|---|
| x² = 5x − 4 | x² − 5x + 4 = 0 | 1, −5, 4 |
| 3x² + 7 = 2x | 3x² − 2x + 7 = 0 | 3, −2, 7 |
| x² = 9 | x² + 0x − 9 = 0 | 1, 0, −9 |
In x² = 9 there is no x term, so b = 0, not "nothing". Writing the zero in keeps the formula from losing its place.
x = −b ± √b² − 4ac2a
2x² − 5x + 3 = 0
a = 2, b = −5, c = 3
b² − 4ac = 25 − 24 = 1
discriminant first
x = 5 ± 14
−b is +5
x = 1.5 or x = 1
two branches
With b = −5, the formula's −b becomes +5. Writing −(−5) out in full is worth the extra second; this is where most formula errors live.
Computing D first tells you what kind of answer to expect, and saves you from hunting for factors that do not exist.
x² − 6x + 2 = 0
D = 36 − 8 = 28
x = 6 ± √282
√28 = 2√7
= 6 ± 2√72
factor out 2
= 3 ± √7
≈ 5.65 or 0.35
Cancelling the 2 requires factoring it out of both terms on top. 6 + 2√72 is 3 + √7, not 3 + 2√7 — the 2 divides the whole numerator.
sum = −ba, product = ca
2x² − 5x + 3 = 0, roots 1 and 1.5
verify them
sum should be 52 = 2.5
1 + 1.5 = 2.5 ✓
product should be 32 = 1.5
1 × 1.5 = 1.5 ✓
Two quick sums confirm both roots at once — faster than substituting each one back into the original equation.
Here the roots are known (or their kind is), and a coefficient is the unknown.
For which k does x² − 10x + k = 0 have exactly one root?
b² − 4ac = 0
one root means D = 0
100 − 4(1)(k) = 0
a = 1, b = −10, c = k
k = 25
check: x² − 10x + 25 = (x − 5)² ✓
For two real roots we would solve D > 0 instead (k < 25), and for none, D < 0 (k > 25).
One root of x² + 4x + k = 0 is 3. Find k and the other root.
3² + 4(3) + k = 0
a root makes the equation true
21 + k = 0, so k = −21
solve for k
3 + r = −4, so r = −7
sum of roots = −ba = −4
3 × (−7) = −21 ✓
product = ca agrees
Build the equation whose roots are 3 and −5.
sum = −2, product = −15
from the two roots
x² − (sum)x + (product) = 0
with a = 1
x² + 2x − 15 = 0
check: (x − 3)(x + 5) ✓
Writing x² + (sum)x + (product) gives x² − 2x − 15 = 0, whose roots are 5 and −3 — the right numbers with the wrong signs. Because the sum of the roots is −ba, the middle coefficient is minus the sum.
The same formula; each needs one tidy-up before a, b and c are read.
A negative leading coefficient: −x² + 4x + 1 = 0
x² − 4x − 1 = 0
multiply every term by −1
D = 16 + 4 = 20, √20 = 2√5
a = 1, b = −4, c = −1
x = 4 ± 2√52 = 2 ± √5
sum 4 = −ba ✓
Different: a positive a keeps every sign easier to track; the roots do not change.
Fraction coefficients: (12)x² + x − 4 = 0
x² + 2x − 8 = 0
multiply every term by 2
D = 4 + 32 = 36
a perfect square
x = −2 ± 62 = 2 or −4
product −8 = ca ✓
Different: clearing fractions first keeps the formula free of halves.
Not in standard form: x(x − 3) = 5
x² − 3x − 5 = 0
expand, then move the 5
D = 9 + 20 = 29
not a perfect square
x = 3 ± √292
irrational roots
Different: x = 5 or x − 3 = 5 is wrong — a product equal to 5, not 0, says nothing about either factor.
Expect to read a, b, c from a rearranged equation, to classify roots by the discriminant, to solve with the formula, or to find a k that gives one root. The fastest route is D first: it tells us whether to factor, use the formula or stop. The 10-second check: the roots must add to −ba and multiply to ca.
Solve
Solve 3x² + 5x − 2 = 0.
a = 3, b = 5, c = −2. The discriminant is 25 + 24 = 49, so x = −5 ± 76, giving 13 and −2. Check with the sum: −ba = −53, and 13 + (−2) = −53 ✓ The product should be ca = −23, and (13)(−2) = −23 ✓ A has both signs reversed.
Hint: c is negative, so −4ac is positive and the discriminant grows. Then check your two roots against −ba before choosing.
