You visited us 3 times! Enjoying our articles? Unlock Full Access!

Vertex

α , β are the...

Question

$α, β$ are the roots of the equation $x^{2} + b x + c = 0$ and $α + 2, β + 2$ are the roots of the equation $x^{2} + p x + q = 0$ . If the minimum value of the expression $x^{2} + b x + c$ is $2$ , find the minimum value of $x^{2} + p x + q$

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2.0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2.00

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The graph of $x^{2}$ + bx + c gets shifted by 2 units when the roots become $α + 2$ and $β + 2$ . The minimum value will remain the same.

For example:
The graph for $x^{2} - 5 x + 6$ and $x^{2} - 11 x + 30$ . The roots are $2, 3$ and $5, 6$ respectively.
The roots of the second equation ( $x^{2} - 11 x + 30)$ are obtained by adding $3$ to the roots of first equation. You can see in the graph that the minimum value is same for both the functions.

Suggest Corrections

Similar questions

Q. If

α & β

are the roots of equation.

x^{2} + p x + q = 0

, then -

- \frac{1}{α}, - \frac{1}{β}

are the roots of the equation.

Q. If α, β are the roots of the equation

x^{2} + p x + q = 0 then - \frac{1}{α} + \frac{1}{β}

are the roots of the equation
(a)

x^{2} - p x + q = 0

(b)

x^{2} + p x + q = 0

(c)

q x^{2} + p x + 1 = 0

(d)

q x^{2} - p x + 1 = 0

$α, β$ are the roots of the equation $x^{2} + b x + c = 0$ and $α + 2, β + 2$ are the roots of the equation $x^{2} + p x + q = 0$ . If the minimum value of the expression $x^{2} + b x + c$ is $2$ , find the minimum value of $x^{2} + p x + q$

Q. If

α, β

are roots of the equation

x^{2} + p x - q = 0

and

γ, δ

are roots of

x^{2} + p x + r = 0,

then the value of

(α - γ) (α - δ)

is-

Q. If

α

and

β

are the roots of the equation

x^{2} + p x + q = 0

, then the equation whose roots are

α^{2} + α β

and

β^{2} + α β

Join BYJU'S Learning Program

α,β are the roots of the equation x2+bx+c=0 and α+2,β+2 are the roots of the equation x2+px+q=0. If the minimum value of the expression x2+bx+c is 2, find the minimum value of x2+px+q

$α, β$ are the roots of the equation $x^{2} + b x + c = 0$ and $α + 2, β + 2$ are the roots of the equation $x^{2} + p x + q = 0$ . If the minimum value of the expression $x^{2} + b x + c$ is $2$ , find the minimum value of $x^{2} + p x + q$