# 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.

## Exampley = x^{2} - 2x, y = x + 4

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

So the right sides of both functions are equal.

x^{2} - 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

x^{2} - 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

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.

These are the graphs of

the quadratic function [y = x^{2} - 2x]

and the line [y = x + 4].

By solving the system,

you found the intersecting points:

(4, 8) and (-1, 3).

## FormulaDiscriminant

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.

## Exampley = x^{2} - x, y = x + k

Find the quadratic equation

from these functions.

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

So the right sides of both functions are equal.

x^{2} - x = x + k

Move x + k to the right side.

Then x^{2} - 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.

1x^{2} - 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.