Simultaneous Equations with One Quadratic
Solve a linear and a quadratic equation together by substituting the linear one into the quadratic, and read the answers as the points where a line meets a curve.
What a learner can do afterwards
- Substitute the linear equation into the quadratic and collect terms to one side
- Give the answers as coordinate pairs, not just x values
- Say what a single repeated solution means about the line and the curve
1 · Read
You can now sketch a curve and see where a line would meet it. This stop finds those meeting points exactly. A line and a curve meet where both equations are true at once, so the method is substitution: take y from the linear equation, put it into the quadratic, and solve. The x values you get are the crossing points, and the number of them matches what your sketch would show: two, one, or none.
The answers must be coordinate pairs, because a solution is a point where the line and the curve meet, and a point needs both an x and a y. Finding x from the quadratic is only half the job: put each x back into the linear equation to get its y. If the quadratic has a repeated root, the line and curve meet at exactly one point, and the line is a tangent to the curve. That is the discriminant-zero case from before, now with a picture.
Solve y = x squared - 2 and y = x + 4. Substitute the linear into the quadratic: x squared - 2 = x + 4. Collect everything to one side: x squared - x - 6 = 0. Factorise: (x - 3)(x + 2) = 0, so x = 3 or x = -2. Now find each y from the linear equation: when x = 3, y = 7, and when x = -2, y = 2. The line meets the curve at (3, 7) and (-2, 2).
Now a line that just touches. Solve y = x squared and y = 2x - 1. Substituting: x squared = 2x - 1, so x squared - 2x + 1 = 0. That factorises to (x - 1) squared = 0, so x = 1 is a repeated root. There is only one meeting point. When x = 1, y = 2(1) - 1 = 1, so the line touches the curve at (1, 1). The repeated root is the algebraic version of a tangent.
And a line that misses. Solve y = x squared + 2 and y = x + 1. Substituting: x squared + 2 = x + 1, so x squared - x + 1 = 0. The discriminant is 1 - 4, which is -3, negative, so this quadratic has no real roots. The line never meets the curve. On a sketch, the curve sits entirely above the line. The discriminant tells you this before you try to factorise, which is useful when the factorisation never comes.
Check your points on both equations, not just the one you solved. A point that works in the quadratic but not in the line is not a meeting point. And when a question asks where a line touches a curve, write 'repeated root' or 'discriminant zero' early, because that is the idea the question is testing.
Substitute the linear equation into the quadratic, collect to one side and solve, then find each y from the linear equation: two roots give two points, a repeated root gives one tangent point, and a negative discriminant gives none.
2 · Watch
3 · Play
Slide the line across the curve: two crossings, one touch, or a clean miss.
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.