How this instrument works
The substitution method solves two equations together by taking the expression one equation gives for a variable and dropping it directly into the other equation in its place, collapsing two unknowns down to one. Here the two equations are a parabola, y = ax² + bx + c, and a line, y = mx + k. Since both describe y, the substitution is immediate: wherever the two graphs meet, ax² + bx + c must equal mx + k at that shared x, so one expression simply replaces y in the other.
Rearranging that equality collects everything on one side into a single quadratic, ax² + (b−m)x + (c−k) = 0, and the quadratic formula then hands back every x where the two graphs actually cross. Because a quadratic can carry zero, one, or two real roots depending on its discriminant, a line can miss a parabola entirely, graze it at exactly one tangent point, or cut cleanly through it at two separate locations — a genuinely richer set of outcomes than two straight lines ever produce.
That richness is exactly what sets this page apart from the Intersection of Two Lines and Systems of Equations pages elsewhere on this site. Both of those solve for where two straight lines cross, and a linear-linear pair always lands on exactly one shared point unless the lines happen to run parallel. Swapping one of those lines for a parabola changes the substitution's output from a linear equation to a quadratic one, which is why two crossing points, one, or none are all genuinely possible outcomes here rather than edge cases to guard against.
The one-point case deserves a closer look, since it's easy to mistake for an error. When the discriminant (b−m)² − 4a(c−k) works out to exactly zero, the line is tangent to the parabola — it touches at a single location without passing through — and the quadratic formula returns the same x value twice rather than failing. A negative discriminant is the true miss: no real x satisfies the equality, meaning the line and the parabola never meet at all across the entire plane.
- Enter the parabola's coefficients into Parabola: a (from y=ax²+bx+c), Parabola: b, and Parabola: c.
- Enter the line's slope into Line: slope (m, from y=mx+k) and its y-intercept into Line: y-intercept (k).
- Read Intersection: x (larger root) and Intersection: x (smaller root) for the shared x-values.
- If both fields return the same figure, the line is tangent to the parabola at that single point.
- A flagged invalid input means the discriminant is negative — this particular line never meets this particular parabola.
Worked example — y=x²−2x−3 meeting y=x−1
The parabola y = x² − 2x − 3 (so a=1, b=−2, c=−3) and the line y = x − 1 (so m=1, k=−1) go into their fields as given. Substituting the line's expression in for y gives x² − 2x − 3 = x − 1, and moving everything to one side leaves x² − 3x − 2 = 0, since b−m = −2−1 = −3 and c−k = −3−(−1) = −2.
Applying the quadratic formula, x = (−(−3) ± √((−3)² − 4(1)(−2))) ⁄ (2·1) = (3 ± √(9+8)) ⁄ 2 = (3 ± √17) ⁄ 2. Since √17 ≈ 4.1231056256, that gives x ≈ 3.5615528128088303 for Intersection: x (larger root) and x ≈ −0.5615528128088303 for Intersection: x (smaller root) — the two exact points where this line cuts through this parabola.
Contrast that with the parabola y = x² meeting the flat line y = 0: substitution gives x² = 0, a discriminant of exactly zero, and a single repeated root at x = 0. The line doesn't cross the curve there at all — it just touches the parabola's own lowest point and lifts away on both sides, the tangent case this same formula handles without any separate rule.
Questions
What does substitution mean in this context?
It means taking the expression one equation gives for a shared variable, here y, and inserting it directly into the other equation in place of that variable. Since the parabola gives y = ax²+bx+c and the line gives y = mx+k, substitution sets ax²+bx+c equal to mx+k, turning two equations in two unknowns into a single quadratic equation in one unknown, x.
How is this different from finding where two straight lines cross?
Two straight lines produce a linear equation after substitution, which has exactly one solution unless the lines are parallel — the Intersection of Two Lines and Systems of Equations pages on this site cover that case. Swapping one line for a parabola turns the same substitution step into a quadratic instead of a linear equation, opening up two crossing points, one tangent point, or no crossing at all, rather than the single guaranteed answer two lines give.
Why can a line and a parabola meet at two points, one point, or none?
Because the substitution collapses to a quadratic equation, and a quadratic's discriminant, (b−m)²−4a(c−k), can be positive, zero, or negative. A positive discriminant gives two distinct real roots and two crossing points; zero gives one repeated root, meaning the line is tangent; negative gives no real root at all, meaning the line and parabola never touch.
What does it mean when the line is tangent to the parabola?
It means the line touches the parabola at exactly one location without passing through to the other side, which happens precisely when the discriminant equals zero. Both Intersection: x (larger root) and Intersection: x (smaller root) come back identical in that case, since the quadratic formula produces the same repeated value from both the plus and minus branches.
What if the discriminant comes out negative?
That means the line never meets the parabola anywhere on the plane — no real value of x makes the substituted equation true. This calculator flags that input as having no real solution rather than returning a pair of intersection points, since the square root of a negative discriminant isn't a real number.