Question

25. Find the value of k and the roots if the difference
between roots of the equation: X2-5X+k -4=0 is 5.
A) K=1, Roots 0 and 1 B) K=4, Roots 0 and 5
C) K=4, Root 5
D) None of these.

Answer 1

EXPLANATION.

α and β are the roots of the equation.

If difference between the roots of equation is,

⇒ x² - 5x + k - 4 = 0 is 5.

As we know that,

Sum of the zeroes of the quadratic polynomial.

⇒ α + β = - b/a.

⇒ α + β = -(-5)/1 = 5.

Products of the zeroes of the quadratic polynomial.

⇒ αβ = c/a.

⇒ αβ = (k - 4)/1 = k - 4.

To find value of k.

Difference of the roots : |α - β| = 5.

Squaring on both sides of the equation, we get.

⇒ (α - β)² = (5)².

⇒ (α + β)² - 4αβ = 25.

Put the values in the equation, we get.

⇒ (5)² - 4(k - 4) = 25.

⇒ 25 - 4k + 16 = 25.

⇒ - 4k + 16 = 0.

⇒ - 4k = - 16.

⇒ k = 4.

Put the value of k = 4 in the equation, we get.

⇒ x² - 5x + k - 4 = 0.

⇒ x² - 5x + 4 - 4 = 0.

⇒ x² - 5x + 0 = 0.

⇒ x² - 5x = 0.

⇒ x(x - 5) = 0.

⇒ x = 0  and  x = 5.

Values of k = 4  and Roots = 0 and 5.

Option [B] is correct answer.

Answer 2

Answer:

Given :-

• The difference between the roots of the equation is x² - 5x + k - 4 = 0 is 5.

To Find :-

• What is the value of k and the roots.

Solution :-

Given Equation :

$\mapsto \bf x^2 - 5x + k - 4 =\: 0$

By comparing with ax² + bx + c = 0 we get,

✫ a = 1

✫ b = - 5

✫ c = k - 4

Now, we have to find the zeroes of the quadratic equations :

☆ In case of sum of the zeroes :

As we know that :

$\footnotesize \bigstar \: \: \sf\boxed{\bold{\pink{Sum\: Of\: Zeroes\: (\alpha + \beta) =\: \dfrac{- b}{a}}}}\: \: \: \bigstar$

So, according to the question by using the formula we get,

$\implies \sf Sum\: of\: Zeroes =\: \dfrac{- (- 5)}{1}$

$\implies \sf Sum\: of\: Zeroes =\: \dfrac{5}{1}$

$\implies \sf\bold{\blue{Sum\: of\: Zeroes =\: 5}}$

☆ In case of product of the zeroes :

As we know that :

$\footnotesize \bigstar \: \: \sf\boxed{\bold{\pink{Product\: of\: Zeroes\: (\alpha\beta) =\: \dfrac{c}{a}}}}\: \: \: \bigstar$

So, according to the question by using the formula we get,

$\implies \sf Product\: of\: Zeroes =\: \dfrac{k - 4}{1}$

$\implies \sf\bold{\blue{Product\: of\: Zeroes =\: k - 4}}$

Now, we have to find the value of k :

$\bigstar$ The difference between the roots of the equation is 5.

So,

$\implies \sf \alpha - \beta =\: 5$

By squaring the both sides we get,

$\implies \sf (\alpha + \beta)^2 - 4\alpha\beta =\: 25$

We have :

• Sum of the Zeroes = 5
• Product of the Zeroes = k - 4

By putting the values we get,

$\implies \sf (5)^2 - 4(k - 4) =\: 25$

$\implies \sf 25 - 4k + 16 =\: 25$

$\implies \sf - 4k + 16 =\: 25 - 25$

$\implies \sf - 4k + 16 =\: 0$

$\implies \sf {\cancel{-}} 4k =\: {\cancel{-}} 16$

$\implies \sf 4k =\: 16$

$\implies \sf k =\: \dfrac{\cancel{16}}{\cancel{4}}$

$\implies \sf k =\: \dfrac{4}{1}$

$\implies \sf\bold{\purple{k =\: 4}}$

Now, we have to find the roots :

$\implies \sf x^2 - 5x + k - 4 =\: 0$

$\implies \sf x^2 - 5x + 4 - 4 =\: 0$

$\implies \sf x^2 - 5x =\: 0$

$\implies \sf x(x - 5) =\: 0$

$\implies \sf x =\: 0\: or\: x - 5 =\: 0$

$\implies \sf\bold{\red{x =\: 0\: or\: x =\: 5}}$

$\therefore$ The value of k is 4 and the roots are 0 and 5 .

Hence, the correct options is option no (B) k = 4, Roots 0 and 5 .