Question

alpha and beta are the zeros of the quadratic polynomial f(x)=ax²+bx+c then evaluate... alpha-beta..class 10​

Answer 1

Step-by-step explanation:

alpha= -b+√b^2-4ac/2a

beta= -b-√b^2 -4ac/2a

Answer 2

Answer:

Given:-

• α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c

To Do:-

• Evaluate:- alpha - beta (or) α - β

Solution:-

Given polynomial,

f(x) = ax² + bx + c ━━━━ (1)

Since, α and β are the zeroes of given polynomial.

So,

• $α+β \:=\: \dfrac{−b}{a}$

• $αβ\:= \: \dfrac{c}{a}$

Therefore,

⇒ (α+β)² = α² + β² + 2αβ

⇒ $(\dfrac{−b}{a})²$ = α² + β² + $\dfrac{c}{a}$

⇒ α² + β² = $\dfrac{b²}{a²}$ - $\dfrac{c}{a}$

Now,

⇒ (α-β)² = α² - β² + 2αβ

⇒ (α-β)² = $\dfrac{b²}{a²}$ - $\dfrac{c}{a}$ + $2\dfrac{c}{a}$

⇒ (α-β)² = $\dfrac{b²}{a²}$ - $\dfrac{c}{a}$

⇒ α-β = $\sqrt{\dfrac{b²}{a²} - \dfrac{c}{a}}$

Hence, the value of α - β is

${\bold{\sqrt{\dfrac{b²}{a²} - \dfrac{c}{a}}}}$

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