If x = \(\frac{2}3\) and x = -3 are the roots of the quadratic equat

1.	If x = \(\frac{2}3\) and x = -3 are the roots of the quadratic equation ax2 + 2ax + 5x + 10 then find the value of a and b.
Answer» Given: ax² + 7x + b = 0 Since, x = \(\frac{2}3\) is the root of the above quadratic equation Hence, it will satisfy the above equation. Therefore, we will get a (\(\frac{2}3\))² + 7 (\(\frac{2}3\)) + b = 0 ⇒ \(\frac{4}9\)a + \(\frac{14}3\) + b = 0 ⇒ 4a + 42 + 9b = 0 ⇒ 4a + 9b = – 42 …(1) Since, x = –3 is the root of the above quadratic equation Hence, It will satisfy the above equation. Therefore, we will get a (–3)² + 7 (–3) + b = 0 ⇒ 9a – 21 + b = 0 ⇒ 9a + b = 21 …..(2) From (1) and (2), we get a = 3, b = – 6

If x = \(\frac{2}3\) and x = -3 are the roots of the quadratic equation ax2 + 2ax + 5x + 10 then find the value of a and b.

Answer»

Given:

ax² + 7x + b = 0

Since,

x = \(\frac{2}3\) is the root of the above quadratic equation

Hence, it will satisfy the above equation.

Therefore, we will get

a (\(\frac{2}3\))² + 7 (\(\frac{2}3\)) + b = 0

⇒ \(\frac{4}9\)a + \(\frac{14}3\) + b = 0

⇒ 4a + 42 + 9b = 0

⇒ 4a + 9b = – 42 …(1)

Since, x = –3 is the root of the above quadratic equation

Hence, It will satisfy the above equation.

Therefore, we will get

a (–3)² + 7 (–3) + b = 0

⇒ 9a – 21 + b = 0

⇒ 9a + b = 21 …..(2)

From (1) and (2), we get

a = 3, b = – 6