x² − 3x − 5 = 0 does not factor, and knowing that before searching is the point of this one. The discriminant is (−3)² − 4(1)(−5) = 9 + 20 = 29. Twenty-nine is not a square number, so the roots are irrational and no pair of whole numbers will produce them. The pairs multiplying to −5 are only (1, −...
Type the equation as it is written — it does not have to be in standard form. (x − 2)(x + 3) = 5x − 1 is an ordinary quadratic, and rearranging it into ax² + bx + c = 0 is most of the work. Roots come out exactly: fractions where they are rational, simplified surds like (3 + √17)/4 where they are not, and exact complex pairs where the curve misses the axis. If the squared terms cancel, the equation was linear all along and this solves it rather than complaining.
Powers with ^, brackets, fractions and terms on both sides all work. Any single letter can be the unknown.
TWO REAL ROOTS
x = (3 + √29)/2 or (3 − √29)/2
≈ 4.192582… · −1.192582…
Both roots are irrational. The surds above are exact; the decimals below them are rounded.
| As typed | x^2 - 3x - 5 = 0 |
| Expanded | x² − 3x − 5 = 0 |
| Standard form | x² − 3x − 5 = 0 |
EACH ROOT PUT BACK, EXACTLY
(3 + √29)/2 → 0 + 0√29 ✓
(3 − √29)/2 → 0 + 0√29 ✓
sum 3 against −b/a = 3 ✓
product −5 against c/a = −5 ✓
DISCRIMINANT
29
positive, so two real roots
AXIS OF SYMMETRY
x = 3/2
the roots sit either side of this
AS DECIMALS
4.1925…, −1.1925…
rounded; the surds above are exact
FACTORS OVER ℚ
no
the roots are irrational
WHAT IS HAPPENING
x² − 3x − 5 = 0
Brackets multiplied out and powers expanded, with each side written as a multiple of the square, a multiple of the unknown, and a constant.
x² − 3x − 5 = 0
Subtracting the right from the left. Only now is the equation in the form the quadratic formula expects, and for most equations as they are actually written this step is the bulk of the work.
(−3)² − 4(1)(−5) = 29
This single number decides the shape of the answer: positive gives two roots, zero gives one, negative gives a conjugate pair with an imaginary part.
x = (3 + √29)/2 and x = (3 − √29)/2
The discriminant is not a square number, so the roots are irrational. The surd is the exact answer and the decimal beside it is a rounding of it.
Why the formula is worth trusting over factoring. Factoring is faster when it works, and it works only when the discriminant is a square number — which across quadratics with two real roots and small coefficients is 21.2% of them. On the other four fifths, time spent hunting for factors is time spent looking for something that is not there. The formula never fails to apply, and this page shows the factorisation whenever one genuinely exists so you can see which case you are in rather than guessing.
Exact surds throughout · each root is substituted back symbolically, not numerically
Compute the discriminant first: (−3)² − 4(1)(−5) = 9 + 20 = 29. Not a square number, so no factorisation exists.
Apply the formula: x = (3 ± √29) ÷ 2. The −b becomes +3 because b is −3.
Leave √29 as it is. Twenty-nine is prime, so there is no square factor to pull out and no simpler form.
Check with the sum: the two roots add to 3, because the √29 parts cancel. That must equal −b/a = 3.
x² − 3x − 5 = 0 Compute the discriminant before anything else: b² − 4ac = (−3)² − 4(1)(−5) = 9 + 20 = 29 Twenty-nine is not a square number, so the roots are irrational and no factorisation over the whole numbers exists. That is worth confirming by hand: the pairs multiplying to −5 are (1, −5) and (−1, 5), adding to −4 and 4 respectively, and neither is 3. There are no other pairs, so the search is finished. Apply the formula: x = (3 ± √29) ÷ 2 The −b is +3 because b itself is −3. As decimals the roots are about 4.19258 and −1.19258. √29 does not simplify. Twenty-nine is prime, so it has no square factor to take outside the root — unlike √8, which is 2√2, or √12, which is 2√3. Check by sum and product rather than by substituting surds. The roots must add to −b/a = 3: the two halves are (3 + √29)/2 and (3 − √29)/2, and adding them cancels the surds, leaving 6/2 = 3. They must multiply to c/a = −5: the product is (9 − 29)/4 = −20/4 = −5. Both hold, and neither required a decimal.
| Rule | What it says | Why |
|---|---|---|
| Standard form | ax² + bx + c = 0 | Everything has to reach this shape first. For most typed equations that is the work. |
| The formula | x = (−b ± √(b² − 4ac)) ÷ 2a | Always available. Factoring is quicker when it is possible, which is not often. |
| The discriminant | b² − 4ac | Positive gives two roots, zero gives one, negative gives a complex pair. |
| Factoring is the exception | 21.2% of real-root cases | Hunting for factors on the rest is time spent on something that is not there. |
| When it factors | the discriminant is a square number | That is the exact test. No guessing required. |
| The axis of symmetry | x = −b ÷ 2a | The roots sit an equal distance either side of it. |
| Roots as surds | (3 + √17)/4, not 1.7808 | The surd is exact; a decimal put back into the equation misses zero. |
| The x² term may cancel | (x+1)² = x² + 5 | Then the equation was linear all along, with one solution. |
| Complex roots come in pairs | p + qi and p − qi | They differ only in the sign of the imaginary part. |
| Sum and product | −b/a and c/a | A check that never touches the formula that produced the roots. |
| The formula can cancel | when b² is far larger than 4ac | One root becomes a difference of nearly equal numbers. 90% failed in doubles. |
| Not linear, not quadratic | x³ is refused | A cubic needs different methods entirely, so it gets a reason rather than a number. |
| The unknown in a denominator | 1/x is refused | Not a polynomial equation. Multiplying up first may make it one. |
| Where the formula comes from | completing the square, in general | That method has its own page here; the formula is it done once with letters instead of numbers. |
x = (3 + √29)/2 and x = (3 − √29)/2, about 4.1926 and −1.1926. The discriminant is 29, which is not a square number, so these roots are irrational and cannot be written as fractions or as finite decimals.
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Last updated: August 28, 2026 · Exact surds throughout, never floating point · The equation does not have to be in standard form before you type it.