System of Equations: Quadratic, Linear

How to solve the system of quadratic and linear equations and find the number of intersecting points: formula, 2 examples, and their solutions.

Example 1

Example

Solution

Both functions are in [y = ...] form.

So the right sides of both functions are equal.

x2 - 2x = x + 4

Substitution Method

The solution is quite similar to
the system of circle and linear equations.

Move x + 4 to the right side.

Factor the right side
x2 - 3x - 4.

Find a pair of numbers
whose product is the constant term -4
and whose sum is the coefficient of the middle term -3.

-4⋅1 = -4
-4 + 1 = -3

Then (x - 4)(x + 1) = 0.

Factor a Quadratic Trinomial

Solve the quadratic equation.

Case 1) x - 4 = 0
Then x = 4.

To find the y value for this case,
put x = 4
into the linear equation y = x + 4.

Then y = 4 + 4 = 8.

x = 4
y = 8

So (4, 8) is the answer for case 1.

Case 2) x + 1 = 0
Then x = -1.

To find the y value for this case,
put x = -1
into the linear equation y = x + 4.

Then y = -1 + 4 = 3.

x = -1
y = 3

So (-1, 3) is the answer for case 2.

Case 1) (4, 8)
Case 2) (-1, 3)

Write these two points.

So (4, 8), (-1, 3) is the answer.

Graph

These are the graphs of
the quadratic function [y = x2 - 2x]
and the line [y = x + 4].

By solving the system,
you found the intersecting points:
(4, 8) and (-1, 3).

Number of Intersecting Points

Formula

Just like the above example,
when solving
a system of quadratic and linear equations,
you'll get a quadratic equation.

From the quadratic equation,
you can find the discriminant D.

This D determines
the number of the intersecting points.

If D is plus,
then they meet at two points.

If D = 0,
then they meet at one point.

If D is minus,
then they do not meet.

Example 2

Example

Solution

Find the quadratic equation
from these functions.

Both functions are in [y = ...] form.

So the right sides of both functions are equal.

x2 - x = x + k

Move x + k to the right side.

Then x2 - 2x - k = 0.

This is the quadratic equation
you're looking for.

It says
find the range of k:
not the roots.

So find the discriminant D.

1x2 - 2x - k = 0.

a = 1
b = -2
c = -k

Then D = (-2)2 - 4⋅1⋅(-k).

(-2)2 = 4
-4⋅1⋅(-k) = +4k

D = 4 + 4k

It says
the given functions have to intersect.

Then they have to meet
at least one point:
one point (D = 0) or two points (D > 0).

So D ≥ 0.

So 4 + 4k ≥ 0.

Move 4 to the right side.

Divide both sides by 4.

4 is plus.
So dividing both sides by 4
does not change the order of the inequality sign.

Then k ≥ -1.

Linear Inequality (One Variable)

So k ≥ -1 is the answer.