One of the roots of the equation x^2 + ax + 72 = 0 is 9. Find the other root and the coefficient a.

For x = 9, the equation x ^ 2 + a * x + 72 = 0 holds. In other words, its left side becomes equal to zero. Knowing this, in order to determine what the coefficient a is equal to, it is enough to substitute instead of x one of the roots of this equation, which is known to us by the condition (x = 9), and then – to express and calculate a:

x ^ 2 + a * x + 72 = 0;
9 ^ 2 + a * 9 + 72 = 0;
81 + 9 * a + 72 = 0;
153 + 9 * a = 0;
a = (- 153): 9 = – 17.

Then, by Vieta’s theorem, the 2nd root is 8.

Answer: coefficient a = – 17, x2 = 8.