Question

Given 3 is one of the roots of the equation x² – px + 2p + 1 = 0, find:

(a) the value of p,
(b) the other root of the equation.​

Answer 1

(a) 3² - 3p + 2p + 1 = 0

9 - p + 1 = 0

10 - p = 0

p = 10

(b) x² - 10p + 21 = 0

(x - 3)(x - 7) = 0

x = 3, 7

The other root of this equation is 7.

Answer 2

To solve the equation x² - px + 2p + 1 = 0 when 3 is a root, substitute 3 into the equation to find p = 10. Then, use the sum of roots formula to find the other root, which is 7.

Step-by-step explanation:

Since 3 is one of the roots of the equation x² – px + 2p + 1 = 0, by the Factor Theorem, we can substitute x = 3 into the equation to find the value of p.

Substituting 3 into the equation:

3² – p(3) + 2p + 1 = 0

9 – 3p + 2p + 1 = 0

10 – p = 0

This simplifies to p = 10. Now that we know p, we can use it to find the other root of the equation. The sum of the roots of a quadratic equation ax² + bx + c = 0 is given by –b/a. In our case, a = 1, and b is –p, thus –(-p)/1, which simplifies to p. As we have one root, 3, the other root will be p – 3, which is 10 – 3 = 7.

Conclusion:

(a) The value of p is 10.

(b) The other root of the equation is 7