A stunt pilot is testing a new plane. The equation that models his height over time is f(x)=15x^2 -195x +950, where x is his the time in seconds and (fx) is his height in metres. Determine when the pilot is below 500 metres.

I subbed in 500 for f(x)

f(x)=15x^2 -195x +950
500 = 15x^2 -195x + 950
0=15x^2 -195x + 950 - 500
0=15x^2 - 195x + 450

What do i do next?

yes - just solve for x the way you normally do, either by grouping as you say, or by the quadratic formula

seems ok to me. From t=3 to t=10 the plane is below 500 m.

at the vertex, t=13/2, f(6.5)= 316.25
f(2.9) = f(10.1) = 510.65

what bothered you about the answer? you expected to get two values for x.

Nevermind

Thank you for clarifications!

Now you can solve for x to find when the pilot was at 500m.

Think of the graph. It's a parabola, opening up. So, the vertex is between the two roots, and below the line y=500.

So, the height will be below 500 for all x between the roots of your equation as shown.

So do I have factor out the 15x^2 - 195x + 450, which will give me (x-?)(x-?)

I did factoring and got this:

f(x)=15x^2 -195x +950
500 = 15x^2 -195x + 950
0=15x^2 -195x + 950 - 500
0=15x^2 - 195x + 450
0=15(x^2 - 13x + 30)
0=15(x-10)(x-3)

x=10 and x=3

It doesn't seem right~

To determine when the pilot is below 500 meters, you need to solve the equation 0=15x^2 - 195x + 450.

Next, you can either factor or use the quadratic formula to solve the equation. Let's go with factoring.

To factor the quadratic equation, we need to find two numbers whose product is 15*450 (6750) and whose sum is -195.

Let's break down 6750 into pairs of factors:

1 * 6750
2 * 3375
3 * 2250
...
50 * 135
75 * 90

If we look at the pair -75 and -120, their sum is -195, which matches the coefficient of the x term in the equation.

So, we can rewrite the equation as:

0 = 15x^2 - 75x - 120x + 450

Next, we group the terms and factor by grouping:

0 = 15x(x - 5) - 120(x - 5)

Now, we can factor out the common factor (x - 5):

0 = (x - 5)(15x - 120)

Setting each factor equal to zero, we have two possible solutions:

x - 5 = 0 => x = 5

15x - 120 = 0 => 15x = 120 => x = 8

Therefore, the pilot is below 500 meters at x = 5 seconds and x = 8 seconds.