We have two functions f and g, defined as:
f(x) = √(5x + 25) - 1
g(x) = x² - 2x - 3
We need to find the points of intersection. These are defined as the values taken by x that make the two functions equal:
f(x) = g(x)
From the definition above:
√(5x + 25) - 1 = x² - 2x - 3
√(5x + 25) = x² - 2x - 2
Now, we take the square on both sides:
5x + 25 = ( x² - 2x - 2 )²
But:
( x² - 2x - 2 )² = x⁴ + 4x² + 4 - 4x³ - 2x² - 2x² + 4x + 4x = x⁴ - 4x³ + 8x + 4
Replacing this:
5x + 25 = x⁴ - 4x³ + 8x + 4
0 = x⁴ - 4x³ + 3x - 21
If we solve the equation, we obtain only two real results:
x1 = -1.6618
x2 = 4.1231
To the nearest tenth:
x1 = -1.7
x2 = 4.1