If the equations x² + 3x + 2 = 0 and ax² + bx + c = 0 have both roots in common, then which of the following is NOT true?

Correct option is C

Given:

The two quadratic equations are:

​$$x^2 + 3x + 2 = 0$$​​

​$$ax^2 + bx + c = 0$$​​

Concept Used:

If two quadratic equations have the same roots, then the ratios of the corresponding coefficients of the two equations must be equal. That is:

​$$\frac{a}{p} = \frac{b}{q} = \frac{c}{r}$$​​

where p, q, r are the coefficients of the first equation, and a, b, c are the coefficients of the second equation.

Solution: 

In this case, for the two equations x² + 3x + 2 = 0 and ax² + bx + c = 0, we have:

p = 1, q = 3, r = 2 (from x² + 3x + 2 = 0).

Therefore, $$\frac{a}{1} = \frac{b}{3} = \frac{c}{2}$$​, which gives us the relationships:

b = 3a,  c = 2a

Now, 

Option A:

4a - 2b + c = 0

Substitute b = 3a and c = 2a:

4a - 2(3a) + 2a = 0

4a - 6a + 2a = 0

0 = 0 (This is true.)

Option B:

a - b + c = 0

Substitute b = 3a and c = 2a:

a - 3a + 2a = 0

0 = 0(This is true)

Option C:

4a = 2c + 2b

Substitute b = 3a and c = 2a:

4a = 2(2a) + 2(3a)

4a = 4a + 6a

4a = 10a (This is NOT true because 4a ≠ 10a)

Option D:

a + c = b

Substitute b = 3a and c = 2a:

a + 2a = 3a

3a = 3a(This is true)

Thus, The statement that is NOT true is Option C: 4a = 2c + 2b.