# Quadratic-Linear System

How to solve the quadratic-linear system and use the discriminant to find the number of the roots: examples and their solutions.

## Example 1: Solve *y* = *x*^{2} - 2*x*, *y* = *x* + 4

Both equations have the same *y*.

So *x*^{2} - 2*x* = *x* + 4.

Move *x* + 4 to the left side.

Then *x*^{2} - 3*x* - 4 = 0.

Factor *x*^{2} - 3*x* - 4.

Factor a quadratic trinomial

Find a pair of numbers

whose product is the constant term [-4]

and whose sum is the middle term's coefficient [-3].

The constant term is (-).

So the signs of the numbers are different:

one is (+), and the other is (-).

(-1, 4) are not the right numbers.

[-4] = -4⋅1

-4 + 1 = [-3]

So -4 and 1 are the right numbers.

Use -4 and +1

to write a factored form:

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

Solve (*x* - 4)(*x* + 1) = 0.

1) *x* - 4 = 0

So *x* = 4.

Solving a quadratic equation by factoring

*x* = 4

Put this into [*y* = *x* + 4].

(You can also put this into [*y* = *x*^{2} - 2*x*].)

Then *y* = 8.

Substitution method

*x* = 4*y* = 8

So, for case 1,

(*x*, *y*) = (4, 8).

See case 2 of (*x* - 4)(*x* + 1) = 0.

2) *x* + 1 = 0

So *x* = -1.

*x* = -1

Put this into [*y* = *x* + 4].

Then *y* = 3.

*x* = -1*y* = 3

So, for case 2,

(*x*, *y*) = (-1, 3).

So (*x*, *y*) = (-1, 3), (4, 8).

Let's see what the answer means.

Below are the graphs of the system:*y* = *x*^{2} - 2*x**y* = *x* + 4.

As you can see,

the solution of the system

are the intersecting points of the functions:

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

## Example 2

Both functions have the same *y*.

So *x*^{2} - *x* = *x* + *k*.

Move *x* + *k* to the left side.

Then *x*^{2} - 2*x* - *k* = 0.

Recall that

the discriminant of a quaratic equation

determines the number of the roots.

The discriminant

If the given functions intersect,

there would be at least one intersecting point.

Then there would be at least one *x*.

So set *D* ≥ 0.*a* = 1*b* = -2*c* = -*k*

Then *D* = (-2)^{2} - 4⋅1⋅(-*k*).

And this is greater than or equal to 0.

(-2)^{2} = 4

-4⋅1⋅(-*k*) = +4*k*

Move 4 to the right side.

Then 4*k* ≥ -4.

Divide both sides by 4.

Then *k* ≥ -1.

This is the answer.

Let's see the relationship

between the discriminant *D*

and the number of the intersecting points.

Below are the graphs of the given functions:*y* = *x*^{2} - *x**y* = *x* + *k*.

If *D* > 0,

there are two roots.

So there are two intersecting points.

If *D* = 0,

there's one root.

So there's one intersecting point:

[*y* = *x* + *k*] touches [*y* = *x*^{2} - *x*].

If *D* < 0,

there's no root.

So there's no intersecting point.