Question

If alpha and beta are the roots of equation x² - 5x + 6 = 0, then find the value of alpha - beta.​

Answer 1

$\underline{\underline{\sf{{\maltese\:\:Given}}}}$

• $\sf\alpha + \beta$are the roots of the equation x² - 5x + 6 = 0.

$\underline{\underline{\sf{\maltese\:\:To\: Find}}}$

• $\sf \: \alpha - \beta = \: ?$

$\underline{\underline{\sf{\maltese\:Calculations \:}}}$

$\bf\red{STEP \:1:-}$

• Compare x² + 5x - 6 = 0 with ax² + bx + c = 0 to get
• a = 1, b = 5 and c = 6.

$\bf\red{STEP \:2:-}$

• Sum of roots $\sf \: \alpha + \beta = \frac{ - b}{a} = \frac{ - ( - 5)}{1}$

or,

• $\sf\alpha + \beta = 5$

$\bf\red{STEP \:3:-}$

• Product of roots $\sf \: \alpha \beta = \frac{c}{a} = \frac{6}{1}$

or,

• $\sf \: \alpha \beta = 6$

$\bf\red{STEP \:4:-}$

$\alpha - \beta = ± \sqrt{ {( \alpha + \beta) }^{2} - 4 \alpha \beta }$

substituting the values,

$\longrightarrow \alpha - \beta = ± \sqrt{ {5}^{2} - 4 \times 6 }$

$\longrightarrow \alpha - \beta = ± \sqrt{25 - 24}$

$\longrightarrow \alpha - \beta = ± \sqrt{1}$

$\purple{\alpha - \beta = 1}$

and

$\pink{\alpha - \beta = - 1}$

HENCE, the value of $\alpha - \beta$ is -1 and +1

Answer 2

$\huge{\tt{\fcolorbox{aqua}{azure}{\color{red}{Answer}}}}$

x2−5x+6=0…(i)

If α and β are two solution of ax2+bx+c=0 then α+β=−ba and αβ=ca.

So from equation (i) and using above we can write α+β=−(−5)=5 and αβ=6 .

Now α+β=5

Or, (α+β)2=52

Or, (α−β)2+4αβ=−(−5)=25

Or, (α−β)2+4×6=25

Or, (α−β)2+24=25

Or, (α−β)2=25−24=1

Or, α−β=± 1(Answer)