Question

1.How can completing the square aid in solving quadratic functions?
2. What indicators predict that a quadratic function will have a complex solution?

Answer 1

1. It helps you find the maximum/minimum at a certain x-value. Thus turning point.

Answer 2

1. Because you can transform a quadratic form ( with x^2) into 2 binomial forms (ax + b) that you can easily solve for x.
2. When the Discriminant D = b^2 - 4ac is negative.
Reminder;
D = b^2 - 4ac > 0: there are 2 real roots
D = 0: double root at x = -b/2a
D < 0: no real roots, there are complex roots.
Reminder.
When D is a perfect square, the quadratic equation can be factored.

Answer 3

1)

When you have a quadratic equation like
$$
y=ax^2+bx+c
$$
Where a,b and c are constants, you can transform this equation from of sum of products to a product of sums, which is what you do when you complete the square, you will have:
$$
y=(x-d)(x-e)
$$

Where d and e are constants, usually different from a, b and c.

This transformation is called completing the square.

Now say you wanted to find out where the quadratic equation crosses the x-axis. You simply set y=0 and find your two solutions:

$$
y=0=(x-d)(x-e)\implies x=d\text{ or } x=e
$$
Two solutions if \(d\ne e\), one solution otherwise.

This is one way that completing the square can help you solve a quadratic function.

2)
As @thu1935 described, when b^2 - 4ac is negative, the solutions (i.e. \(x\)) will have complex solutions.