Solving Quadric Equations
Problem 1
x2 - 2x - 15 = 0
We have a = 1, b = -2, c = -15.
To solve this equation by factoring, we have to find such m and n that m * n = c = -15, and m + n =b = -2.
There are 4 pairs that have product equal to -15 => (15, -1), (-15, 1), (5 -3), (-5, 3).
For (15,-1) , we have m + n = 15 - 1 = 14
For (-15, 1), we have -15 + 1 = -14
For (5, -3) , we have 5 - 3 = 2
For (-5, 3), we have 3 - 5 = -2 = b
Thus, our m = -5, and our n = 3
We can rewrite our equation in a form
(x - 5)(x + 3) = 0
Really, (x - 5)(x + 3) = x2 -5x + 3x - 15 = x2 - 2x - 15
To solve this equation for x , we have to solve two equations
x - 5 = 0 and x + 3 = 0
For the first equation x = 5. For the second equation x = -3.
Problem 2
2h2 + 4h + 1 = 0
We have a = 2, b = 4, c = 1
Our first step is to find a discriminant.
D = b2 - 4ac = 42 - 4 * 2 * 1 = 16 - 8 = 8
Then we find a square root from the discriminant.
√D = √8 ≈ 2.82843
After that we can use the quadratic formula
h = (-b ±√D)/2a = (-4 ±2.82843)/4
Finally, we obtain h1 = -1.707, h2 = -0.293
The same result can be received by the completing the square method.
2h2 + 4h + 1 = 0 may be presented as 2(h2 + 2h) = -1 => h2 + 2h = -1/2 => h2 + 2h +1 - 1 = -1/2 =>
=> h2 + 2h + 1= 1/2 => (h + 1)2 = 0.5 => h + 1 = ±√0.5 => h = -1 ± √0.5
The final result is h1 = -1 - 0.707 = -1.707, h2 = -1 + 0.707 = -0.293
Although we used different methods, our roots are the …
