Δ positive and a perfect square
- Input
- a=2, b=3, c=-2
- Expected output
- Δ = 25
Since 25=5², the square root is exact and both roots, 0.5 and −2, come out rational, with no infinite decimals.
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calculate discriminant quadratic equation
The discriminant Δ = b²−4ac determines the nature of roots without computing the square root. It is the first step of the quadratic formula.
Since 25=5², the square root is exact and both roots, 0.5 and −2, come out rational, with no infinite decimals.
The parabola's vertex touches the x-axis at x=−3; the equation has a single repeated real root there.
A negative Δ means the parabola never crosses the x-axis; the equation has no real root, only two complex conjugate roots.
The discriminant Δ = b²−4ac determines the nature of roots: Δ>0 → two distinct real roots; Δ=0 → double root; Δ<0 → no real roots.
It has no real solution: the parabola never crosses the x-axis. The equation still has two solutions, but they only exist among complex numbers, in the form a±bi.
If a, b and c are integers and Δ is a perfect square (like 25, 36 or 49), the square root of Δ is exact and both roots come out rational; if Δ is not a perfect square, the roots are irrational numbers.
Yes: −Δ/(4a) is the parabola's vertex height on the y-axis. When Δ=0, that height is zero, meaning the vertex sits exactly on the x-axis, which explains why the double root occurs right there.
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