Name: Condition for Common Roots of Quadratic Equation
Uploaded: 2019-07-19
Duration: 36 min 41 s
Description: In this video, we will learn the condition for one common root and the condition for both roots common for two given quadratic equations with the help of exa...

Question

If

α

and

β

are the two distinct roots of

x^{2} + 2 (k - 3) x + 9 = 0

, then find the values of

k

such that

α, β \in (- 6, 1)

.

Accepted Answer

Both roots of quadratic equation $x^{2} + 2 (k - 3) x + 9 = 0$ lie between $(- 6, 1)$ .

As we know conditions for quadratic equation $a x^{2} + b x + c = 0, a > 0$ has roots between two number $d$ and $e$ are $D \geq 0, a \cdot f (d) > 0, a \cdot f (e) > 0 & d < - \frac{b}{2 a} < e$ .

$(i)$ $D \geq 0$

$\Rightarrow 4 {(k - 3)}^{2} - 4 \times 9 \geq 0$

$\Rightarrow {(k - 3)}^{2} - 9 \geq 0$

$\Rightarrow k^{2} - 6 k + 9 - 9 \geq 0 \Rightarrow k^{2} - 6 k \geq 0$

$\Rightarrow k (k - 6) \geq 0$

$\Rightarrow k \in (- \infty, 0] \cup [6, \infty) . . . . . (i)$

$(i i)$ $a . f (d) > 0 \Rightarrow 1 \cdot f (- 6) > 0$

$\Rightarrow 1 . (36 - 12 k + 36 + 9) > 0$

$\Rightarrow 12 k < 81 \Rightarrow k < \frac{27}{4}$

$\Rightarrow k \in (- \infty, \frac{27}{4}) . . . . . (i i)$

$(i i i)$ $a \cdot f (e) > 0 \Rightarrow 1 \cdot f (1) > 0$

$\Rightarrow 1 + 2 k - 6 + 9 > 0$

$\Rightarrow 2 k + 4 > 0 \Rightarrow k > - 2$

$\Rightarrow k \in (- 2, \infty) . . . . . (i i i)$

$(i v)$ $d < - \frac{b}{2 a} < e \Rightarrow - 6 < - \frac{2 (k - 3)}{2} < 1$

$\Rightarrow - 6 < - k + 3 < 1$

$\Rightarrow - 9 < - k < - 2$

$\Rightarrow 2 < k < 9$

$\Rightarrow k \in (2, 9) . . . . . (i v)$

Intersection of $(i), (i i), (i i i)$ and $(i v)$ will be,

$\Rightarrow k \in (6, \frac{27}{4})$ .

If $α$ and $β$ are the two distinct roots of $x^{2} + 2 (k - 3) x + 9 = 0$ , then find the values of $k$ such that $α, β \in (- 6, 1)$ .

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If α and β are the two distinct roots of x2+2k-3x+9=0, then find the values of k such that α,β∈-6,1.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If $α$ and $β$ are the two distinct roots of $x^{2} + 2 (k - 3) x + 9 = 0$ , then find the values of $k$ such that $α, β \in (- 6, 1)$ .