Determine the number of solutions

Question

y=3x^2

y=3x-2

Help!

Answer 1

Okay, Sophia, let's see if we can0 figure this out...

First equation: y=3x^2

Second equation: y=3x-2

If the first equation AND the second equation both equal y (on the left side), then the first equation AND the second equation are equal to one another. So now we have 3x^2=3x-2. Graphically, this indicates that we are looking for the intersection (if there is one) of a parabola and a line.

Now, let's subtract 3x-2 from both sides of this equation, which will give us 3x^2-(3x-2)=3x^2-3x+2=0

This is a quadratic equation, which can be solved using the quadratic formula: For ax^2+bx+c=0, we have

x={-b+-sqrt[b^2-4ac]}/2a (here +- means "plus or minus" and sqrt means "square root")

For our quadratic equation, a=3, b=-3 and c=2. Substituting values for a, b and c, we have

x={-(-3)+-sqrt[(-3)^2-4(3)(2)]}/2(3), or x={3+-sqrt[9-24]}/6

Given that a,b and c are real numbers, there are three possibilities:

b^2-4ac is positive, in which case the roots are real and unequal

b^2-4ac is zero, in which case the roots are real and equal

b^2-4ac is negative, in which case the roots are imaginary and unequal.

Notice that b^2-4ac is 9-24, which is -15 and negative; therefore, there are TWO SOLUTIONS to this equation, both of which are imaginary, and the two solutions are unequal; or, put a different way, there are no real solutions to this problem (because the solutions are imaginary).

I hope this helps.

Answer 2

One equation is a quadfratic, the other is a straight line. The "solutions" are the points where their graphs intersect. At those points, the two equations are equal. So set them equal:

3x² = 3x - 2

3x² - 3x + 2 = 0

Use the discriminant (b² - 4ac) to determine the number of real solutions:

1. If the discriminant > 0, there are two solutions (the line crosses the parabola twice)
2. If the discriminant = 0, there is one solution (the line touches the parabola tangentially just once)
3. If the discriminant < 0, there are no real solutions (the line does not cross the parabola)