a) x^2 - 8x + 15 = 0.
Here we multiply 15 and the coefficient of x^2 = 15*1 = 15.
So now we think of two numbers whose product is 15 and its sum gives us the middle coefficient term -8.
15 = -3 * -5.
And -3 + -5 = -8. We replace the -8 with this expression.
x^2 - 8x + 15 = 0.
x^2 -3x - 5x + 15 = 0. Factorize now.
x (x - 3) - 5(x - 3) = 0.
(x -3)(x - 5) = 0.
(x-3) = 0 or (x -5) = 0.
x = 0 +3, x = 0 +5. x = 3, or 5.
b) x^2 + 11x + 28 = 0.
Following similar steps like above:
28 = 14(2), 28 = 7(4).
But it is only 7 + 4 = 11. Which gives the middle coefficient. So we adopt 7(4).
x^2 + 11x + 28 = 0
x^2 + 4x + 7x + 28 = 0. Factorize.
x(x + 4) + 7(x + 4) = 0.
(x + 4)(x +7) = 0.
x+4 = 0 or x+7 = 0.
x = 0-4 or x = 0 -7. x = -4 or -7.
Cheers.