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Solve any quadratic equation with every step written out, exactly.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
A quadratic equation is any equation you can write as ax² + bx + c = 0, where a is not zero. The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, solves every single one of them - which is unusual and worth appreciating. Factoring works only when the numbers happen to be friendly; completing the square works always but takes practice; the formula works always and takes only care.
The part doing the real work is under the square root. b² - 4ac is called the discriminant, and its sign decides the shape of the answer before you calculate anything else: positive means two roots, zero means one repeated root, negative means no real solution at all. Checking the discriminant first tells you what kind of answer to expect, which is the fastest way to catch an arithmetic slip later.
The formula is not a single move, and that is the most common misunderstanding about it. It is a short procedure: substitute, evaluate the discriminant, take the root, split the ±, then reduce each branch. This calculator walks all five so you can see which one you would have got wrong.
Type a, b and c into the boxes. Whole numbers, decimals and fractions like 1/2 all work. A missing term means a coefficient of zero: for x² - 4 = 0, b is 0.
Type something like 2x^2 - 5x + 2 = 0 into the equation field and press Read it. Terms on both sides are collected onto the left automatically, so 3x^2 = 2x + 1 works too.
Step four gives you b² - 4ac. Note its sign and predict the shape of the answer - two roots, one, or none - before scrolling on.
If your answer disagreed, the useful question is not which number is right but which step diverged. Find that line and the mistake explains itself.
The formula is not an axiom. It is what you get by completing the square on the general equation once, so that nobody has to do it again on their own numbers. Working through it a single time is the difference between remembering the formula and knowing it.
a x^2 + b x + c = 0
a cannot be zero - without an x² term there is nothing quadratic to solve, and the 2a underneath would be zero too.
x^2 + (b/a) x + c/a = 0
Completing the square needs a bare x², so the a has to come off the front. This is also the step that puts a into the denominator, where it stays for the rest of the derivation.
x^2 + (b/a) x = -c/a
x^2 + (b/a) x + b^2/(4a^2) = -c/a + b^2/(4a^2)
Half of b/a is b/2a, and its square is b²/4a². This is the move the whole derivation turns on: it is exactly what the left side was missing to be a perfect square.
(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)
There is the discriminant, b² - 4ac, arriving on its own before anyone gave it a name.
x + b/(2a) = ±sqrt(b^2 - 4ac)/(2a)
Both signs, because both square to the same thing. The 2a comes out of √(4a²) as |2a|, and the ± in front absorbs the sign either way.
x = (-b ± sqrt(b^2 - 4ac))/(2a)
Nothing in those seven lines used a particular a, b or c, which is why the result holds for every quadratic there is. It is also why the discriminant decides the answer's shape: it is the number under the root, and if it is negative, step six has nowhere to go.
The value of b² - 4ac decides everything about the answer's shape before you take a single square root.
| Discriminant | Real roots | The graph | Example |
|---|---|---|---|
| b² - 4ac > 0 | Two different real roots | Crosses the x-axis twice | x² - 5x + 6 = 0 → x = 3, x = 2 |
| b² - 4ac = 0 | One repeated root | Touches the x-axis once | x² - 4x + 4 = 0 → x = 2 |
| b² - 4ac < 0 | No real solution | Never reaches the x-axis | x² + 2x + 5 = 0 |
| A perfect square | Two rational roots | Crosses at two fractions or integers | 2x² - 5x + 2 = 0 → x = 2, x = 1/2 |
| Positive, not a square | Two irrational roots | Crosses at two radicals | x² - 4x + 1 = 0 → x = 2 ± √3 |
a = 1, b = -5, c = 6The discriminant is (-5)² - 4(1)(6) = 25 - 24 = 1. Its square root is exactly 1, so the roots are (5 + 1)/2 = 3 and (5 - 1)/2 = 2. Because the discriminant is a perfect square, both roots are rational and the equation also factors as (x - 3)(x - 2) = 0. This is the case where factoring would have been quicker.
a = 2, b = -5, c = 2The discriminant is 25 - 16 = 9, so √9 = 3 and the roots are (5 + 3)/4 = 2 and (5 - 3)/4 = 1/2. Notice that the second root stays as 1/2 and is never shown as 0.5 - a fraction is the exact value, and the decimal is an approximation of it that happens to be short.
a = 1, b = -4, c = 1The discriminant is 16 - 4 = 12, which is not a perfect square. √12 simplifies to 2√3, so the roots are (4 ± 2√3)/2 = 2 ± √3. This is where most calculators start printing 3.7320508 and stop being useful: 2 + √3 is the answer, and any decimal is a rounding of it.
a = 1, b = 2, c = 5The discriminant is 4 - 20 = -16. No real number squares to a negative, so the equation has no real solution and the derivation stops there. The parabola sits entirely above the x-axis. If your course has reached complex numbers the roots are -1 ± 2i, but whether that counts as an answer depends on which number system you have been taught to work in.
