Quadratic Formula Calculator
Your details
Result
Roots
- Discriminant (b² − 4ac)
- 49 = 5² − 4(2)(-3)
- What the discriminant tells you
- Positive — two distinct real roots
- Vertex
- (-1.25, -6.125)
- Axis of symmetry
- x = -1.25
- Parabola opens
- upward — vertex is the minimum
- Factored form
- 2(x − 0.5)(x + 3)
- y-intercept
- (0, -3)
- The discriminant is 49, which is positive, so the parabola crosses the x-axis twice and there are two distinct real roots. Checking the discriminant first tells you this before any root is calculated.
- 49 is a perfect square (7²), which is why the roots come out rational. That also means the equation could have been factored directly without the formula.
- The vertex sits at x = −b/2a = -1.25, giving y = -6.125. Since a is positive, the parabola opens upward and this is the minimum.
The formula, substituted
Each stage of x = (−b ± √(b² − 4ac)) / 2a with your coefficients in place.
| Stage | Value |
|---|---|
| b² | 25 |
| 4ac | -24 |
| b² − 4ac (discriminant) | 49 |
| √(b² − 4ac) | 7 |
| −b | -5 |
| 2a | 4 |
What the discriminant means
| b² − 4ac | Roots | Graph |
|---|---|---|
| Positive | Two distinct real | Crosses the x-axis twice |
| Zero | One repeated real | Touches the x-axis at the vertex |
| Negative | Two complex conjugates | Never reaches the x-axis |
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a, which solves any equation of the form ax² + bx + c = 0. The part under the square root, b² − 4ac, is the discriminant and decides how many real roots exist.
x = (−b ± √(b² − 4ac)) / 2a vertex at x = −b/2a- a
- Coefficient of x²; cannot be zero, and its sign sets the opening direction
- b
- Coefficient of x; shifts the parabola sideways
- c
- The constant, which is also the y-intercept
- b² − 4ac
- The discriminant — how many real roots there are
The quadratic formula is what you get by completing the square on ax² + bx + c = 0 in general, which is why it works for every quadratic rather than only the factorable ones. The practical habit worth building is to compute the discriminant first, before touching the rest. Its sign answers the question completely: positive means the parabola crosses the x-axis twice and there are two distinct real roots; zero means it touches at a single point, its vertex, giving one repeated root; negative means it never reaches the axis and the roots are complex conjugates. If the discriminant is also a perfect square, the roots are rational and the equation could have been factored directly without the formula at all. The ± is what produces two answers, and the two roots sit symmetrically either side of x = −b/2a, which is both the axis of symmetry and the x-coordinate of the vertex. That symmetry is a useful check: the average of your two roots must equal −b/2a, and their product must equal c/a.
- Write the equation in the form ax² + bx + c = 0.
- Identify a, b and c, keeping their signs.
- Calculate the discriminant b² − 4ac first.
- Check its sign to see how many real roots exist.
- Take the square root and apply the ± for both roots.
- Divide each of the two results by 2a for the roots.
Worked example: Solving 2x² + 5x − 3 = 0
Solving 2x² + 5x − 3 = 0
Here a = 2, b = 5 and c = −3. Start with the discriminant: b² = 25 and 4ac = 4 × 2 × −3 = −24, so b² − 4ac = 25 − (−24) = 49. That sign change is where most errors creep in — a negative c makes the 4ac term negative, so subtracting it adds. The discriminant is positive, so there are two distinct real roots, and 49 is a perfect square, which means they will be rational. The square root is 7, so x = (−5 ± 7) ÷ 4. Taking the plus gives x = 2 ÷ 4 = 0.5, and the minus gives x = −12 ÷ 4 = −3. Both check out: the factored form is 2(x − 0.5)(x + 3), and the symmetry test holds — the roots average to −1.25, which is exactly −b/2a. The vertex therefore sits at (−1.25, −6.125), and since a is positive the parabola opens upward, making that point the minimum.
What each quadratic formula input means
- a — coefficient of x²
- The quadratic coefficient. It cannot be zero — with a = 0 the equation is linear, not quadratic. Its sign decides whether the parabola opens up or down.
- b — coefficient of x
- The linear coefficient. It shifts the parabola sideways and sets the axis of symmetry at −b/2a.
- c — constant term
- The constant. It is also the y-intercept, since substituting x = 0 leaves only c.
What this quadratic formula does not account for
- a cannot be zero. With a = 0 the x² term disappears and the equation is linear, solved by x = −c/b instead.
- A negative c makes the 4ac term negative, so subtracting it adds to the discriminant. This sign handling is the most common source of error.
- A negative discriminant means no real roots. In a word problem about a physical quantity, that usually indicates the described situation cannot occur.
- When the discriminant is a perfect square the roots are rational and the equation factors directly — the formula still works but is more effort than needed.
- The equation must be rearranged to equal zero before identifying coefficients. Reading a, b and c off an equation with terms on both sides gives the wrong answer.
- Two useful checks: the roots must average to −b/2a, and their product must equal c/a. Both catch arithmetic slips immediately.
- Irrational roots are best left in surd form for exact work. Decimal approximations accumulate error if fed into further calculations.
Frequently asked questions
What is the discriminant?
The expression b² − 4ac under the square root. Its sign tells you the number of real roots before you calculate any: positive gives two, zero gives one repeated, and negative gives none.
How do you solve 2x² + 5x − 3 = 0?
The discriminant is 25 − (−24) = 49, whose square root is 7. So x = (−5 ± 7) ÷ 4, giving x = 0.5 and x = −3. It factors as 2(x − 0.5)(x + 3).
What does it mean when the discriminant is negative?
There are no real roots — the parabola never crosses the x-axis. The two solutions are complex conjugates. In a physical word problem it usually means the scenario described is impossible.
Where does the quadratic formula come from?
From completing the square on ax² + bx + c = 0 in general terms. That is why it works for every quadratic, including those that cannot be factored over the rationals.
How do you find the vertex of a parabola?
The x-coordinate is −b/2a, which is also the axis of symmetry. Substitute it back into the equation for the y-coordinate. For 2x² + 5x − 3 the vertex is at (−1.25, −6.125).
Can a quadratic have only one solution?
Yes, when the discriminant is exactly zero. The two roots coincide at x = −b/2a, and geometrically the parabola touches the x-axis at its vertex rather than crossing it.
Sources and review
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