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Introduction

Equation √5...

Question

Equation $\sqrt{5 x^{2} - 6 x + 8} - \sqrt{5 x^{2} - 6 x - 7} = 1$ has

A

All roots positive

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B

All roots negative

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C

One root positive and the other negative

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D

Two root positive and one root negative

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Solution

The correct option is C One root positive and the other negative
$\sqrt{5 x^{2} - 6 x + 8} - \sqrt{5 x^{2} - 6 x - 7} = 1$
squaring both side and we get
$(\sqrt{5 x^{2} - 6 x + 8} - \sqrt{5 x^{2} - 6 x - 7})^{2} = 1^{2}$
$(\sqrt{5 x^{2} - 6 x + 8})^{2} + (\sqrt{5 x^{2} - 6 x - 7})^{2} - 2 x \sqrt{(5 x^{2} - 6 x + 8) (5 x^{2} - 6 x - 7)} = 1$
$\Rightarrow 5 x^{2} - 6 x + 8 + 5 x^{2} - 6 x - 7 - 2 \sqrt{(5 x^{2} - 6 x + 8) (5 x^{2} - 6 x - 7)} = 1$
$\Rightarrow 10 x^{2} - 12 x + 1 - 2 \sqrt{(5 x^{2} - 6 x + 8) (5 x^{2} - 6 x - 7)} = 1$
$\Rightarrow 10 x^{2} - 12 x = 2 \sqrt{(5 x^{2} - 6 x + 8) (5 x^{2} - 6 x - 7)}$
$\Rightarrow 2 (5 x^{2} - 6 x) = 2 \sqrt{(5 x^{2} - 6 x + 8) (5 x^{2} - 6 x - 7)}$
$\Rightarrow (5 x^{2} - 6 x) = \sqrt{15 x^{2} - 6 x + 8) (5 x^{2} - 6 x - 7)}$
On squaring both side and we get
$\Rightarrow (5 x^{2} - 6 x)^{2} = (5 x^{2} - 6 x + 8) (5 x^{2} - 6 x - 7)$
$\Rightarrow 25 x^{4} + 36 x^{2} - 60 x^{3} = 25 x^{4} - 30 x^{3} - 35 x^{2} - 30 x^{3} + 36 x^{2} + 42 x + 40 x^{2} - 48 x - 56$
$\Rightarrow 5 x^{2} - 6 x - 56 = 0$
$5 x^{2} - 6 x - 56 = 0$
$5 x^{2} - (20 - 14) x - 56 = 0$
$5 x^{2} - 20 x + 14 x - 56 = 0$
$5 x (x - 4) + 14 (x - 4) = 0$
$(x - 4) (5 x + 14) = 0$
$\begin{matrix} x - 4 = 0 5 x + 14 = 0 \end{matrix} ∣ ∣ ∣ \begin{matrix} x = 4 x = \frac{- 14}{5} \end{matrix}$
One root position and other negative
This is the answer.

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Similar questions

Q. The sum of the roots of the equation

\sqrt{5 x^{2} - 6 x + 8} - \sqrt{5 x^{2} - 6 x - 7} = 1

Q. Number of rational roots of the equation

\sqrt{5 x^{2} - 6 x + 8} - \sqrt{5 x^{2} - 6 x - 7} = 1

Q. The sum of the roots of the equation

\sqrt{5 x^{2} - 6 x + 8} - \sqrt{5 x^{2} - 6 x - 7} = 1

Q. The roots of the equation 5x² – 6x + 1 = 0 are

समीकरण 5x² – 6x + 1 = 0 के मूल हैं

Solve the equation 5 $x^{2}$ – 6 $x$ – 2 = 0 by the method of completing the square. Find the positive root.

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