Question

How many real solutions does the equation have whose related function is graphed below?

Answer 1

`2`

Explanation:

The real solutions to a quadratic occur where the curve turn at the x axis or crosses the x axis.

These are found either by observing the graph of the quadratic, or, and far more accurate solving the quadratic equation when it is equated to zero.

`ax^2+bx+c=0`

When a the parabola of a quadratic turns at the x axis as in the graph below.

graph{y=(x-3)^2 [-2, 8.89, -2, 2.445]}

The equation has repeated roots ( some people refer to as having only one root, or solution ). When the parabola crosses the x axis as in your example there will be two distinct and real roots.

If the parabola is completely above or below the x axis as in the graphs below, then there are no real solutions.

graph{y=x^2-6x+12 [-10.44, 15, -12.77, 20.24]}

graph{y=-x^2+6x-12 [-5, 10, -12.77, 5]}

So you can always tell from the graph of a quadratic whether it has real or non-real solutions.

Real if it turns at the `x` axis or crosses it.

Non-real if it is completely above or below the `x` axis.

Hope this helps