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 …