Question

Maximum integral value of a forwhich one root of quadratic equation 2ax? - (a+1x -3 =0 is less than unity and other exceeds unity.​

Answer 1

EXPLANATION.

Maximum integral value of a.

One roots of quadratic equation : 2ax² - (a + 1)x - 3 = 0 is less than unity and other exceeds unity.

As we know that,

Conditions of :

One roots should be less than and other greater than x₀.

⇒ F(x) = ax² + bx + c.

⇒ (a) = D ≥ 0.

⇒ (b) = a. f(x₀) > 0.

Equation : 2ax² - (a + 1)x - 3 = 0.

⇒ (a) = D ≥ 0.

⇒ [(a + 1)² - 4(2a)(-3)] ≥ 0.

⇒ [(a + 1)² + 24a] ≥ 0.

⇒ (a² + 1 + 2a) + 24a ≥ 0.

⇒ a² + 1 + 2a + 24a ≥ 0.

⇒ a² + 26a + 1 ≥ 0.

⇒ a = - 26 ± √(26)² - 4(1)(1)/(2).

⇒ a = - 26 ± √676 - 4/2.

⇒ a = - 26 ± √672/(2).

⇒ a = - 26 ± 4√42/(2).

⇒ a = - 13 ± 2√42.

⇒ a = - 13 + 2√42   and   a = - 13 - 2√42.

⇒ a ∈ (- ∞, - 13 - 2√42] ∪ [- 13 + 2√42, ∞). - - - - - (1).

(b) = a. f(1) > 0.

⇒ 2a [2a(1)² - (a + 1)(1) - 3] > 0.

⇒ 2a[2a - a - 1 - 3] > 0.

⇒ 2a[a - 4] > 0.

⇒ a ∈ (- ∞, 0) ∪ (4, ∞). - - - - - (2).

From equation (1) and (2), we get.

(1) ∪ (2).

⇒ a ∈ (- ∞, - 13 - 2√42] ∪ [4, ∞).

Answer 2

Answer:

Given :-

• Maximum integral value for which one root of quadratic equation 2ax?-(a+1x-3=0 is than unity and other exceeds unity. )

□To find :-

• Maximum integral value for quadratic equation.

♧Explanation :-

• Here given,
• $F(x) = a {x}^{2} + bx + c$
• $(a) = D \geqslant 0$
• $(b) = a.f(x) > 0$

1. Here equation is given ,

• $2a {x}^{2} - (a + 1)x - 3 = 0$
• $(a) = d \geqslant 0$
• $(( {a + 1}^{2} ) - 4(2a)( - 3)) \geqslant 0$
• $(( {a + 1}^{2} ) + 24)) \geqslant 0$
• $( {a}^{2} + 1 + 2a) + 24a \geqslant 0$
• ${a}^{2} + 1 + 2a + 24a \geqslant 0$
• ${a}^{2} + 26a + 1 \geqslant 0$
• $a = - 26 \binom{ + }{ - } \sqrt{ {26}^{2} } - 4(1) \frac{1}{2}$

• $a = - 26 \binom{ + }{ - } \sqrt{ \frac{676}{2} }$
• $a = - 26 \binom{ + }{ - } 4 \sqrt{ \frac{42}{2} }$
• $a = - 13 \binom{ + }{ - } 2 \sqrt{42}$ (i)
• $here \: a \: value \: is \: given$
• $(b) = a.f(1) > 0$
• $2a(2a ({1}^{2} ) - (a + 1)(1) - 3) > 0$
• $2a(2a - a - 1 - 3) > 0$
• $2a(a - 4) > 0$
• a belongs to
• $(- \infty ,0) \: and \:$ and U (4, inifinity) (2)

♧Here from equation 1 and 2 we get the answer as

• (1)U (2)

Conclusion :-

• If u got any problem in this answer you can refer the above problem a best moderator answer.

♧Hope it helps u mate .

♧Thank you .