determine the number of solutions of each system of equations

Question

{y=x^2-5x+9

{y=3

these are in one

Answer 1

Note: in the complex number system. There is always at least 1 solution to any quadratic equation.

If a, b, and c are all real numbers, the following is true:

The expression b² - 4ac is called the discriminant.

If its value is zero for a given set of values a, b, and c, then there is one real solution. It is called a double real root. The graph of the expression is tangent (just touches, but does not cross) the x-axis.

If the value of the discriminant is positive, there are two real solutions and the graph crosses the x-axis at two distinct points.

If the value of the discriminant is negative, there are no real solutions. The graph of the expression will not cross the x-axis. There will be two distinct complex solutions of the form d±ei where i=√(-1) and d and e are both real numbers. This is called a complex conjugate pair.

(If any of a, b, or c is complex, then the solutions may be complex and the solution may not be a conjugate pair.)

Complex solutions often arise in electrical engineering problems.

Answer 2

If y = x² – 5x + 9

and y = 3

then you can substitute 3 for y in the first equation to get

3 = x² – 5x + 9

Subtracting 3 from both sides gives

x² – 5x + 6 = 0

The number of solutions can be 0, 1, or 2.

This is determined by calculating b² – 4ac

There are 3 cases:

1. b² – 4ac > 0 means you have 2 solutions
2. b² – 4ac = 0 means you have 1 solution
3. b² – 4ac < 0 means you have 0 solutions

(Technically: In case 3 there are two complex number solutions, but the question is asking for real solutions.)

(Vocabulary lesson: b² – 4ac is called the discriminant.)

Here a = 1, b = –5, and c = 6 and b² – 4ac 5² – 4(1)(6) = 25 – 24 = 1

1 > 0 so you have 2 solutions

If needed, you can find the 2 solutions by factoring

If you can find a pair of numbers a and b where

a × b = 6 and

a + b = –5

then x² – 5x + 6 = (x + a)(x + b)

–2 × –3 = 6 and

–2 × –3 = –5

which means that

x² – 5x + 6 = (x – 2)(x – 3) = 0

and the two solutions are x = 2 and x = 3