justyourcalculatorQuadratic Formula Calculator
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Quadratic Formula Calculator
Solve ax² + bx + c = 0 with exact roots
Roots
x = 1/2, x = -2
x₁ ≈ 0.5, x₂ ≈ -2
- x₁
- 0.5
- x₂
- -2
- Discriminant (b² - 4ac)Positive: two real roots
- 25
- Vertex (h, k)
- (-0.75, -3.125)
- Axis of symmetry
- x = -3/4 ≈ -0.75
- y-intercept
- (0, -2)
- Parabola opens
- Upward (minimum at vertex)
- Sum of roots (-b/a)
- -3/2 ≈ -1.5
- Product of roots (c/a)
- -1
- Vertex form
- y = 2(x + 0.75)² - 3.125
How it was calculated
- 2x² + 3x - 2 = 0
- x = (-b ± √(b² - 4ac)) / (2a)
- Discriminant D = b² - 4ac = 3² - 4 × 2 × (-2) = 25
- D > 0, so there are two real roots.
- x = (-3 ± √25) / 4 = (-3 ± 5) / 4
- x₁ = 0.5, x₂ = -2
- Factored form: 2(x - 0.5)(x + 2)
- Vertex: h = -b / (2a) = -0.75, k = c - b² / (4a) = -3.125
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How the quadratic formula calculator works
Every quadratic equation ax² + bx + c = 0 (with a ≠ 0) is solved by the quadratic formula x = (-b ± √(b² - 4ac)) / (2a). The part under the root, D = b² - 4ac, is the discriminant.
If D is positive there are two different real roots, if it's zero there's one repeated root (the parabola just touches the x-axis), and if it's negative the roots are two complex numbers a ± bi.
When a, b and c are whole numbers or short decimals, the roots are also shown exactly, as simplified fractions or radicals like (-3 ± √17) / 4, which is the form most teachers want. The vertex (h, k) is the turning point of the parabola y = ax² + bx + c.
Using and checking your result
Published by JustYourCalculator. Check the units and assumptions above, and compare a known example before relying on the output. Calculations use browser arithmetic and may round displayed values.
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