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

Let

4 x^{2} - 4 (α - 2) x + α - 2 = 0 (α \in R)

be a quadratic equation. Find the values of

{}^{'}α^{'}

for which

(i)

Both roots are real and distinct.

(ii)

Both roots are equal.

(iii)

Both roots are imaginary

(iv)

Both roots are opposite in sign.

(v)

Both roots are equal in magnitude but opposite in sign.

Accepted Answer

$(i)$ For both the roots to be real and distinct, discriminant $(△)$ must be strictly greater than zero i.e. $△ > 0$ .

(For $a x^{2} + b x + c = 0$ , $△ = b^{2} - 4 a c$ )

$\Rightarrow 16 {(α - 2)}^{2} - 4 (4) (α - 2) > 0$

$\Rightarrow (α - 2) (α - 3) > 0$

$∴ α \in (- \infty, 2) \cup (3, \infty)$

$(ii)$ For both the roots to be equal, the discriminant must be equal to zero i.e. $△ = 0$ .

$\Rightarrow 16 {(α - 2)}^{2} - 4 (4) (α - 2) = 0$

$\Rightarrow (α - 2) (α - 3) = 0$

$∴ α \in \{2, 3\}$ or $α = 2, 3$ .

$(iii)$ For both the roots to be imaginary, the discriminant must be strictly less than zero i.e. $(i)$ .

$\Rightarrow 16 {(α - 2)}^{2} - 4 (4) (α - 2) < 0$

$\Rightarrow (α - 2) (α - 3) < 0$

$∴ α \in (2, 3)$

$(iv)$ For the roots to have opposite sign, product of the roots must be always negative.

(For $a x^{2} + b x + c = 0$ , product of the roots is $\frac{c}{a}$ )

$\Rightarrow \frac{α - 2}{4} < 0$

$\Rightarrow α < 2$

$∴ α \in (- \infty, 2)$

$(v)$ For both roots to have opposite sign and equal in magnitude, the sum of the roots must be zero.

(For $a x^{2} + b x + c = 0$ , sum of the roots is $\frac{- b}{a}$ )

$\Rightarrow \frac{- (- 4 (α - 2))}{4} = 0$

$\Rightarrow (α - 2) = 0$

$∴ α = 2$

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Let 4x2-4α-2x+α-2=0 α∈R be a quadratic equation. Find the values of α'' for which i Both roots are real and distinct. ii Both roots are equal. iii Both roots are imaginary iv Both roots are opposite in sign. v Both roots are equal in magnitude but opposite in sign.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !