Question

Consider the system of quadratic equations

y = 3x^2 - 5x,
y = 2x^2 - x - c,
where c is a real number.

(a) For what value(s) of c will the system have exactly one solution (x,y)?

(b) For what value(s) of c will the system have more than one real solution?

(c) For what value(s) of c will the system have no real solutions?

Solutions to the quadratics are (x,y) pairs. Your answers will be in terms of c, but make sure you address both x and y for each part.

Answer 1

9514 1404 393

Answer:

(a) c = 4

(b) c < 4

(c) c > 4

Explanation:

(a) The solutions to the system of equations can be found by substituting one expression for y into the other equation:

3x^2 -5x = 2x^2 -x -c

x^2 -4x +c = 0 . . . . . . . subtract the right side expression

The discriminant is

d = b^2-4ac = (4)^2 -4(1)(c) = 16-4c.

The system will have exactly one solution when d = 0.

16 -4c = 0

4 -c = 0

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(b) There will be more than one real solution when d > 0

4 -c > 0

c < 4 . . . . two real solutions

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(c) There will be no real solutions when d < 0.

4 -c < 0

c > 4 . . . . no real solutions

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Additional comment

The question posed here simply asks for a value of c. It does not ask for the solutions (x, y). We can count them without knowing exactly what they are.