Quadratic Formula Calculator

Enter three numbers, click Calculate. This quadratic formula calculator returns the discriminant and both roots instantly, right on the page.

• Coefficient a (of x²) — the number in front of $x^2$. For the preloaded equation $2x^2 + 8x + 3 = 0$, enter 2.
• Coefficient b (of x) — the number in front of $x$: 8.
• Constant c — the lonely number at the end: 3.
• Press Calculate. Three lines come back: the discriminant $b^2 - 4ac = 40$, then the two roots $x_1 \approx -0.4189$ and $x_2 \approx -3.5811$.

One habit before typing: rewrite the equation with zero alone on one side, in the shape $ax^2 + bx + c = 0$. $2x^2 + 8x + 3 = 0$ is already there; an equation like $x^2 = 6x - 9$ must first become $x^2 - 6x + 9 = 0$ before a, b, and c can be read off.

This page is more than a static quadratic formula calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. “Solve two x squared plus eight x plus three equals zero” — the AI runs the formula and explains every substitution step by step.
• Take a photo of your homework. No typing: the AI reads a, b, and c straight off the page and solves step by step.
• Ask about the ideas, not just the number. “Why does the discriminant decide how many roots there are?” The AI explains — and you can check every number against the quadratic formula calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Hand the AI a word problem that hides a quadratic and it builds the equation before solving. Follow-ups are welcome, so ask “why?” as often as you need: this quadratic formula calculator page keeps teaching.

Every quadratic equation $ax^2 + bx + c = 0$ yields to one formula — no guessing, no trial values, no waiting for a factor to appear:

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Read the formula as a recipe: flip the sign of $b$, add or subtract the square root of the discriminant, and divide the whole numerator by $2a$. The $\pm$ is why a quadratic can have two roots — one answer rides the plus, the other the minus.

The formula is not handed down from nowhere: it is what completing the square does to $ax^2 + bx + c = 0$ once, in general, so you never have to do it again for each equation. What matters for homework is simpler — it always works. Every quadratic, every set of coefficients, same one line. The quadratic formula calculator above is that line at one click, and the examples below walk it by hand.

Before touching the formula, compute $D = b^2 - 4ac$. The discriminant alone tells you how many real roots are waiting — which is why the quadratic formula calculator prints it first.

1. D > 0: two different real roots. The square root is a positive number, so the $\pm$ produces two distinct answers. The preloaded $2x^2 + 8x + 3 = 0$ has $D = 40 > 0$: two roots, $x_1 \approx -0.4189$ and $x_2 \approx -3.5811$.
2. D = 0: one repeated root. The square root is zero, plus and minus agree, and both branches land on the same number: $x = -\dfrac{b}{2a}$. The parabola touches the x-axis at exactly one point.
3. D < 0: no real roots. The square root of a negative number is not real, so no real x satisfies the equation. The parabola floats entirely above or below the x-axis. Over the complex numbers two roots exist — but the quadratic formula calculator reports the real story honestly instead of inventing numbers.

Checking the discriminant first is free and saves work: if a classmate's answer to a $D < 0$ equation is a real number, something went wrong before the formula was ever applied. Compute $D$, name the case, then run the formula.

Solve $2x^2 + 8x + 3 = 0$ with the quadratic formula — the values preloaded in the quadratic formula calculator above, so every line below can be checked with one click.

1. Read off the coefficients: $a = 2$, $b = 8$, $c = 3$.
2. Compute the discriminant: $D = b^2 - 4ac = 8^2 - 4(2)(3) = 64 - 24 = 40$. Positive — two different real roots are coming.
3. Take the square root: $\sqrt{40} \approx 6.3249$.
4. Plus branch: $x_1 = \dfrac{-8 + 6.3249}{4} = \dfrac{-1.6751}{4} \approx -0.4189$.
5. Minus branch: $x_2 = \dfrac{-8 - 6.3249}{4} = \dfrac{-14.3249}{4} \approx -3.5811$.

Both roots are negative here, which is worth a second's thought: the parabola $y = 2x^2 + 8x + 3$ opens upward, crosses the x-axis twice to the left of zero, and dips below the axis between the roots.

For an exact answer, stop one step earlier and keep the radical: $x = \dfrac{-8 \pm \sqrt{40}}{4}$. The decimals $-0.4189$ and $-3.5811$ are what the quadratic formula calculator reports, rounded to four places; the radical form is what a textbook answer key usually wants. Type 2, 8, 3 into the calculator above and both forms trace back to the same $D = 40$.

Solve $x^2 - 6x + 9 = 0$. Here $a = 1$, $b = -6$, $c = 9$ — watch the signs: b is negative because the x-term reads minus 6x.

1. Discriminant first: $D = (-6)^2 - 4(1)(9) = 36 - 36 = 0$. The repeated-root case.
2. Formula: $x = \dfrac{-(-6) \pm \sqrt{0}}{2 \times 1} = \dfrac{6 \pm 0}{2}$.
3. Both branches agree: $x = \dfrac{6}{2} = 3$.

So the equation has exactly one solution, $x = 3$, counted twice. The structure behind it: $x^2 - 6x + 9$ is the perfect square $(x - 3)^2$, and the equation says $(x - 3)^2 = 0$ — which pins $x$ to 3 and nothing else. Graphically, the parabola does not cross the x-axis; it touches it at the single point $x = 3$ and bounces off.

A zero discriminant is not a failure — it is information. It tells you the quadratic is a perfect square before you ever try to factor it, and it warns that any classmate reporting two answers here has copied one of them wrong. The quadratic formula calculator above prints $D = 0$ first for exactly this reason: name the case, then believe the roots.

Give factoring ten seconds before reaching for the formula. Some quadratics split on sight:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

Two numbers that multiply to $6$ and add to $-5$ are $-2$ and $-3$, so $x^2 - 5x + 6 = 0$ becomes $(x - 2)(x - 3) = 0$. The zero product property finishes the job: $x = 2$ or $x = 3$. No square roots, no decimals, two lines of work.

When does factoring win? When the discriminant is a perfect square — integer roots are waiting, and factoring finds them whole. When it is not, as with $D = 40$ in Example 1, whole-number factoring is impossible and the formula is the only road.

The honest division of labor: try factoring for ten seconds; reach for the formula the moment it stalls. The formula is the universal fallback — it never fails, never minds fractions, and never cares whether the quadratic was built to factor nicely. The quadratic formula calculator above plays that fallback at one click, and it stays impartial about which route tonight's homework needed.

Four slips the quadratic formula calculator catches every day:

1. Not dividing the whole numerator by $2a$. The fraction bar stretches under $-b \pm \sqrt{b^2 - 4ac}$ as one piece. Dividing only the $-b$ by $2a$ and tacking the square root on afterwards produces a wrong root with a confident face. Draw the bar first, put everything under it, then divide.
2. Dropping the sign of $-b$. With $b = 8$ the numerator starts at $-8$; with $b = -6$ (Example 2) it starts at $+6$. The formula flips the sign of $b$ for you — but only if you let it.
3. Doing the arithmetic out of order. Compute $D$, then $\sqrt{D}$, then the division by $2a$ — the square root comes out before the fraction is evaluated. Reordering those steps is how a correctly copied formula produces an incorrect number.
4. Forgetting to check that $a \neq 0$. No $x^2$ term means no quadratic: $8x + 3 = 0$ is linear, with the single root $x = -\dfrac{3}{8}$, and the formula's $2a$ denominator would be zero anyway. Make the check a reflex.

One habit catches all four: substitute each root back into $ax^2 + bx + c = 0$ and watch both sides agree. The quadratic formula calculator above never skips that check — neither should your notebook.