Question

Alpha beta roots xsqare minus px plus q =0find alpha minus beta

Answer 1

AnSwer:-

The value of $\bold{\tt{\alpha-\beta \ is \ \sqrt{p^{2}-4q^{2}}}}$

Given:-

The given quadratic equation is

${x^{2}-px+q=0}$

Roots are ${\alpha \ and \ \beta}$

To find:-

The value of $\bold{\alpha-\beta}$

SoLution:-

The quadratic equation is ${x^{2}-px+b}$

Here, a=1, b=-p and c=q

${\alpha+\beta=\frac{-b}{a}}$

i.e ${\alpha+\beta=p}$....(1)

${\alpha\beta=\frac{c}{a}}$

i.e. ${\alpha\beta=q}$....(2)

From identity

$\bold{(a-b)^{2}=(a+b)^{2}-4ab}$

From (1) and (2)

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

${(\alpha-\beta)^{2}=p^{2}-4q^{2}}$

On taking square root of both the answer

${\alpha-\beta=\sqrt{p^{2}-4q^{2}}}$

The value of $\bold{\tt{\alpha-\beta \ is \ \sqrt{p^{2}-4q^{2}}}}$

Answer 2

$\color{darkblue}\underline{\underline{\boxed{\star{\sf Corrected \ Question:}}}}$

$\rm{\alpha \ and \ \beta \ are \ the \ roots \ of \ a \ quadratic }$

$\rm{equation \ x^{2}-px+q=0}$

$\rm{Find \ \alpha \ - \ \beta.}$

_________________________________________

$\color{green}\underline{\underline{\boxed{\star{\sf Answer:}}}}$

$\rm\pink{The \ value \ of \ \alpha \ - \ \beta : \ \sqrt{p^{2}-4q^{2}}}$

_________________________________________

$\sf\large\underline\orange{Given:}$

$\rm{The \ given \ quadratic \ equation \ is \ :}$

${x^{2}-px+q=0}$

$\rm{The \ roots \ of \ the \ equation \ are \ \alpha \ and \ \beta}$

_________________________________________

$\sf\large\underline\green{To \ Find:}$

$\rm{The \ value \ of \ \alpha-\beta}$

_________________________________________

$\color{red}\underline{\underline{\boxed{\star{\sf Solution:}}}}$

$\rm{The \ quadratic \ equation \ is \ x^{2}-px+b=0}$

$\rm{We \ have,}$

$\rm{a=1 , \ b=-p, \ c=1 }$

$\rm{We \ Know}$

$\star{\alpha+\beta=\frac{-b}{a}}$

$\implies{\alpha+\beta=p}$.................[a]

$\star{\alpha\beta=\frac{c}{a}}$

$\implies{\alpha\beta=q}$.....................[b]

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

$\rm{................using \ (a-b)^{2}=(a+b)^{2}-4ab}$

$\rm{From \ equations \ [a] \ and \ [b]}$

$\therefore\rm{(\alpha-\beta)^{2}=p^{2}-4q^{2}}$

$\rm{Taking \ Square \ roots \ to \ the \ both \ sides}$

$\rm{We \ get,}$

$\implies{\alpha-\beta=\sqrt{p^{2}-4q^{2}}}$

$\rm{\purple{\boxed{\pink{\star{\bf{The \ value \ of \ \alpha-\beta \ is \ \sqrt{p^{2}-4q^{2}}}}}}}}$

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$\rm\purple{Hope \ it \ Helps \ :))}$