Systems with quadratics (linear-quadratic)
Systems with Quadratics
Systems with Quadratics is a topic in Quadratic Functions in the California Common Core State Standards. It is aligned to Standard A-REI.7, which requires students to solve a simple system of a linear and a quadratic equation in two variables algebraically and graphically.
A linear-quadratic system is solved by substitution; the graphs meet at zero, one, or two points.
Theory
A linear-quadratic system pairs a line with a parabola. Solve by substitution:
- Set the two \(y\)-expressions equal.
- Rearrange into a quadratic \(=0\).
- Solve — the number of real roots is the number of intersection points.
Set equal and solve:
How to solve the system
- Set the expressions equal.
- Move all terms to one side.
- Solve the quadratic (factor or formula).
- Find each \(y\) and write the points.
Set equal and solve.
| \(x^2\) | \(=\) | \(x+2\) |
| \(x^2-x-2\) | \(=\) | \(0\) |
| \((x-2)(x+1)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(2,\ -1\) |
Points \((2,4)\) and \((-1,1)\).
Set equal and solve.
| \(x^2-4\) | \(=\) | \(-x-2\) |
| \(x^2+x-2\) | \(=\) | \(0\) |
| \((x+2)(x-1)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(-2,\ 1\) |
Points \((-2,0)\) and \((1,-3)\).
Set equal.
| \(x^2-2x+1\) | \(=\) | \(0\) |
| \((x-1)^2\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(1\) |
One point \((1,1)\): the line is tangent.
Set equal.
| \(x^2+1\) | \(=\) | \(-1\) |
| \(x^2\) | \(=\) | \(-2\) |
No real solution — the graphs do not meet.
Common pitfalls
Frequently asked questions
How do you solve a linear-quadratic system?
Set the expressions equal and solve the resulting quadratic.
How many solutions can there be?
Zero, one, or two, depending on the discriminant.
What does one solution mean graphically?
The line is tangent to the parabola.
Do you need the \(y\)-values?
Yes — substitute each \(x\) back to get the full point.