Suppose the equation that describes the change in water level (y) with respect to temperature (x): y = x³ - 4x² - 7x + 10
You find the x-intercepts of the cubic equation by finding the roots. Let y be zero:
x³ - 4x² - 7x + 10 = 0
Now, find possible factors of the constant term in the equation: 10. These could be 1, 2, 5, and 10. Suppose we choose 1. We substitute x=1. If the answer is zero, then x=1 is one of the roots.
(1)³ - 4(1)² - 7(1) + 10 = 0
Hence, x=1 or x-1 = 0 is one of the roots. We use this to manipulate the rest of the terms in the original equation, such that you can factor out (x-1). Substitute -x² - 3x² to -4x³ because they are just equal. Same is true for +3x - 10x for -7x:
x³ -x² - 3x²+3x - 10x + 10 = 0
Factor out like terms such that they have a common factor of (x-1) as much as possible.
x³ -x² = x²(x-1)
- 3x²+3x = -3x(x-1)
- 10x + 10 = -10(x-1)
The factored equation is:
x²(x-1)-3x(x-1)-10(x-1) = 0
Factor out (x-1):
(x-1)(x²-3x-10) = 0
Now, we have the quadratic equation left to be solved:x²-3x-10
Use the quadratic formula to find the roots:
x = [-b +/- √(b²-4ac)]/2a,
The a, b and c coefficients are based on the general form of the quadratic equation: ax² + bx + c. Hence, a=1, b=-3 and c=-10. Substituting the values:
x = [-(-3) +/- √((-3)²-4(1)(-10))]/2(1)
x = 5, -2
Therefore, the roots of the cubic equations are x = 1, x = 5 and x = 2. These are the x-intercepts of the equation. At this points, there is no change in the height. The graph is shown in the attached picture