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.
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Answer:
(a) c = 4
(b) c < 4
(c) c > 4
Step-by-step 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.