If the quadratic equation 5x2 - 5k - 4x - 92k-0 has real distinct roots- then which of the following is always true-

The given quadratic equation 5x^2+(5x+4)x+92k=0 has real and distinct roots. ∴ D gt; 0 ⇒ b^2 4ac gt; 0 ⇒ (5k+4)^2 4 × 5 ×92k gt; 0 ⇒ 25k^2+16+40k 90k ...

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Standard X

Mathematics

Discriminant

If the quadra...

Question

If the quadratic equation $5 x^{2} + (5 k + 4) x + \frac{9}{2} k = 0$ has real distinct roots, then which of the following is always true?

\frac{3}{5} < k < \frac{9}{5}

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k < \frac{3}{5}

k > \frac{9}{5}

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k < \frac{2}{5}

k > \frac{8}{5}

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\frac{2}{5} < k < \frac{8}{5}

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Solution

The correct option is C $k < \frac{2}{5}$ or $k > \frac{8}{5}$

The given quadratic equation $5 x^{2} + (5 x + 4) x + \frac{9}{2} k = 0$ has real and distinct roots.
$∴ D > 0$
$\Rightarrow b^{2} - 4 a c > 0$
$\Rightarrow (5 k + 4)^{2} - 4 \times 5 \times \frac{9}{2} k > 0$
$\Rightarrow 25 k^{2} + 16 + 40 k - 90 k > 0$
$\Rightarrow 25 k^{2} - 50 k + 16 > 0$
$\Rightarrow 25 k^{2} - 40 k - 10 k + 16 > 0$
$\Rightarrow 5 k (5 k - 8) - 2 (5 k - 8) > 0$
$\Rightarrow (5 k - 8) (5 k - 2) > 0$
$\Rightarrow k > \frac{8}{5}$ or $k < \frac{2}{5}$
Hence, the correct answer is option (3).

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Q. (1)
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(i)

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(2)
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(ii)

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(3)
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(iii)

2 x^{2} - 6 x + \frac{9}{2} = 0

(4)
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(iv)

3 x^{2} - 4 x + 1 = 0

(5)
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(v)

(x + 4)^{2} - 8 x = 0

(6)
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(vi)

(x - \sqrt{2})^{2} - 2 (x + 1) = 0

(7)
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(vii)

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(8)
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x (1 - x) - 2 = 0

(9)
State whether the following quadratic equation have two distinct real roots. Justify your answer
(ix)

(x - 1) (x + 2) + 2 = 0

(10)
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(x)

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If the quadratic equation 5x2+(5k+4)x+92k=0 has real distinct roots, then which of the following is always true?

If the quadratic equation $5 x^{2} + (5 k + 4) x + \frac{9}{2} k = 0$ has real distinct roots, then which of the following is always true?