# Ideas in Geometry/Instructive examples/Lesson 4: Algebraic Geometry

The **equation of a line** is L(x)=mx+b, where m is the slope and b is the y-intercept.

The **equation of a parabola** is P(x)=ax^2+bx+c.

A line that is **tangent** to a parabola goes through the parabola at exactly one point.

To find P(x)=L(x), we can set P(x)-L(x)=0.

This equation will have 2 roots because the answer will be a quadratic since a quadratic minus a line equals a quadratic of some sort. THESE ROOTS MUST BE THE SAME, so P(x)-L(x)=(x-c)^2.

**Example 1:** Find the line tangent to y=x^2-3x+1 at the point x=3.

**Solution 1:**

P(x)-L(x)=(x-c)^2

x^2-3x+1-L(x)=(x-3)^2

-3x+1-L(x)=-6x+9

L(x)=8-3x

**Example 2:** Find the line tangent to y=x^3-3x^2+4x-1 at the point x=0.

**Solution 2:**

x^3-3x^2+4x-1-L(x)=x^2(x-c)

4x-1-L(x)=0

L(x)=4x-1