The sum of all the roots of the equation | x 2 - 8 x + 15 | - 2 x + 7 = 0 is [2023]

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Solution

Q.
The sum of all the roots of the equation $| x^{2} - 8 x + 15 |$ $- 2 x + 7 = 0$ is [2023]

1 $11 + \sqrt{3}$

2 $11 - \sqrt{3}$

3 $9 + \sqrt{3}$

4 $9 - \sqrt{3}$

Ans.
(3)

Given equation is $| x^{2} - 8 x + 15 |$ $- 2 x + 7 = 0$

$x^{2} - 8 x + 15 \geq 0$

$\Rightarrow (x - 3) (x - 5) \geq 0 \Rightarrow x \leq 3 or x \geq 5$

$When x \leq 3 or x \geq 5$

$x^{2} - 8 x + 15 - 2 x + 7 = 0 \Rightarrow x^{2} - 10 x + 22 = 0$

$∴$ $x = 5 + \sqrt{3}$ $[5 - \sqrt{3} is rejected]$

$When 3 < x < 5$

$- (x^{2} - 8 x + 15) - 2 x + 7 = 0 \Rightarrow x^{2} - 8 x + 15 + 2 x - 7 = 0$

$\Rightarrow x^{2} - 6 x + 8 = 0 \Rightarrow (x - 4) (x - 2) = 0$

$\Rightarrow x = 4 [2 does not lie between 3 and 5, so rejected]$

$Sum of roots = 5 + \sqrt{3} + 4 = 9 + \sqrt{3}$

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