x² − 4 = 0 is the shortest quadratic worth studying, and the one where people most often produce half an answer. Taking the square root of both sides gives x = 2, and stops — but (−2)² is also 4, so x = −2 works just as well. Squaring destroys the sign, and undoing it has to put both possibilities b...
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 = 2 or −2
Both roots are rational, so the quadratic factors: (x − 2)(x + 2) = 0.
| As typed | x^2 - 4 = 0 |
| Expanded | x² − 4 = 0 |
| Standard form | x² − 4 = 0 |
| Factored | (x − 2)(x + 2) = 0 |
EACH ROOT PUT BACK, EXACTLY
2 → 0 ✓
−2 → 0 ✓
sum 0 against −b/a = 0 ✓
product −4 against c/a = −4 ✓
DISCRIMINANT
16
positive, so two real roots
AXIS OF SYMMETRY
x = 0
the roots sit either side of this
AS DECIMALS
2, −2
exact — the roots are rational
FACTORS OVER ℚ
yes
the discriminant is a square number
WHAT IS HAPPENING
x² − 4 = 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 − 4 = 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.
(0)² − 4(1)(−4) = 16
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 = 2 and x = −2
The discriminant came out a square number, so the roots are fractions and the quadratic factors over the rationals.
(x − 2)(x + 2) = 0
A product is zero exactly when one of its factors is zero, so each bracket gives a root directly. Factoring is quicker than the formula when it is available, and it is available precisely when the discriminant is a square number.
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
Recognise the shape: x² − 4 is a difference of two squares, since 4 is 2². That form always factors as (x − 2)(x + 2).
Set each bracket to zero separately. A product is zero exactly when one of its factors is, so x − 2 = 0 or x + 2 = 0.
Read both answers: x = 2 and x = −2. Taking a square root and keeping only the positive value is where the second root goes missing.
Check both in the original: 2² − 4 = 0 and (−2)² − 4 = 0. Both hold, because squaring removes the sign.
x² − 4 = 0 Three routes, all giving the same pair. By square roots. Add 4 to both sides: x² = 4. Now take the square root, remembering that both a positive and a negative number square to 4: x = ±2 By factoring. Four is 2², so this is a difference of two squares: x² − 4 = (x − 2)(x + 2) A product is zero exactly when one of its factors is zero, so x − 2 = 0 gives x = 2 and x + 2 = 0 gives x = −2. By the formula. Here a = 1, b = 0 and c = −4: discriminant: 0² − 4(1)(−4) = 16 x = (0 ± √16) ÷ 2 = ±2 Check both: 2² − 4 = 4 − 4 = 0 and (−2)² − 4 = 4 − 4 = 0. The lost root is the thing to watch. Writing x² = 4 and then x = 2 looks complete and is exactly half an answer. The square root symbol on its own means the positive root only, which is why the ± has to be written in by hand when undoing a square.
| 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 = 2 and x = −2. Both satisfy it, because squaring a negative number gives a positive result: (−2)² is 4, exactly as 2² is. Keeping only the positive answer is the commonest slip on this equation and it loses half the solution.
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Last updated: August 31, 2026 · Exact surds throughout, never floating point · The equation does not have to be in standard form before you type it.