How many real solutions does the function shown on the graph have?

a) no real solutions
b) one real solution
c) two real solutions
d) cannot be determined
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y = (x+2)^2 + 3 is shown on the graph.

I am having trouble figuring this one out. I have been looking in my book for a while now. Somebody please help, it would be very much appreciated!

( x + 2 )² + 2 = 0

Subtract 2 to both sides

( x + 2 )² + 2 - 2 = 0 - 2

( x + 2 )²= - 2

Take square root of both sides

x + 2 = ± √( - 2 )

x + 2 = ± √( - 1 ∙ 2 )

x + 2 = ±√( - 1 ) ∙ √2

x + 2 = ± i √2

Subtract 2 to both sides

x + 2 - 2 = ± i √2 - 2

x = - 2 ± i √2

The solutions are:

x = - 2 + i √2 and x = - 2 - i √2

The function ( x + 2 )² + 2 no real solutions.

Answer a) is correct

Just looking at the equation, we can tell that the vertex is (-2,2) and the parabola opens upwards. So clearly, it cannot cross the x-axis. So clearly, no real solution

I think it could be a) no real solutions?

Am I correct? If not, please explain...

Whoops. I meant the graph was showing: y=(x+2)^2+2

I was thinking that it is a) because it does not have any x-intercepts?

Ah, the graph of (x+2)^2 + 3! Let me put on my comedy hat and help you out with this one.

Well, my dear friend, to determine the number of real solutions, we have to rely on the trusty Discriminant. Now, the Discriminant is like the Yoda of equations - it tells you a lot using just a few symbols.

In this case, since we have a parabola (x+2)^2 + 3, the equation is in the form of ax^2 + bx + c, where a = 1, b = 4, and c = 7. Now, let's calculate the Discriminant using the formula: b^2 - 4ac.

So, plugging in our values, we get (4)^2 - 4(1)(7) = 16 - 28 = -12. Oh dear, that's negative! And as you know, negative Discriminants mean no real solutions, just like my dreams of being a tightrope walker. So the answer is: a) no real solutions.

But hey, don't let that disappoint you. There's always a positive side to everything. And in this case, it's that you don't have to worry about those pesky real solutions. Stay clowny, my friend!

To determine the number of real solutions for the function shown on the graph, you can analyze the graph visually or use algebraic methods.

1. Visual analysis:
Look at the graph of the function y = (x+2)^2 + 3. Notice that the graph is an upward-opening parabola because the coefficient of the x^2 term is positive, and its vertex is at (-2, 3). Since the parabola is opening upward, it means that the graph does not intersect or touch the x-axis. Therefore, there are no real solutions in this case.

The correct answer is a) no real solutions.

2. Algebraic analysis:
To analyze the function algebraically, you can set y = 0 and solve for x. Let's set (x+2)^2 + 3 = 0 and simplify the equation:

(x+2)^2 + 3 = 0
(x+2)^2 = -3

At this point, notice that the square of a real number is always non-negative, which means it can never be negative. Therefore, there are no real solutions to this equation.

Again, the correct answer is a) no real solutions.

So, both the visual and algebraic methods confirm that there are no real solutions for the given function.