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Question Number 183847 by Michaelfaraday last updated on 30/Dec/22

	Answered by MJS_new last updated on 31/Dec/22

	$5 x^{2} - 7 x = t$ $\sqrt{t + 10} - \sqrt{t - 9} = 1$ $\sqrt{100} - \sqrt{81} = 1$ $t = 90$ $x = - \frac{18}{5} \lor x = 5$
	Commented by Michaelfaraday last updated on 31/Dec/22

	$t h a n k s s i r$
	Commented by MJS_new last updated on 31/Dec/22

	$sometimes you need sharp eyes instead of$ $sharp brains$
	Answered by a.lgnaoui last updated on 31/Dec/22

	${\sqrt{5 x^{2} - 7 x + 10} - \sqrt{(5 x^{2} - 7 x + 10) - 19})}^{2} = 1$ $5 x^{2} - 7 x + 10 = z$ $z + (z - 19) - 2 \sqrt{z (z - 19} = 1$ $\sqrt{z (z - 19)} = 10 - z$ $z (z - 19) = 100 + z^{2} - 20 z$ $100 - z = 0 \Rightarrow z = 100$ $5 x^{2} - 7 x - 90 = 0$ $△ = 49 + 20 \times 90 = 1849 = 43^{2}$ $x = \frac{7 \pm 43}{10} x = 5$ $x_{1} = 5$ $v e r i f i c a t i o n$ $\sqrt{125 - 35 + 10} - \sqrt{125 - 35 - 9} = 10 - 9 = 1$
	Commented by Michaelfaraday last updated on 31/Dec/22

	$t h a n k s s i r$
	Answered by mr W last updated on 31/Dec/22

	$a = \sqrt{5 x^{2} - 7 x + 10}$ $b = \sqrt{5 x^{2} - 7 x - 9}$ $a - b = 1$ $a^{2} - b^{2} = 10 - (- 9) = 19$ $(a - b) (a + b) = 19$ $a + b = 19$ $\Rightarrow a = \frac{1 + 19}{2} = 10$ $\sqrt{5 x^{2} - 7 x + 10} = 10$ $5 x^{2} - 7 x + 10 = 10^{2}$ $5 x^{2} - 7 x - 90 = 0$ $\Rightarrow x = \frac{7 \pm 43}{10} = - \frac{18}{5}, 5 ✓$
	Commented by Michaelfaraday last updated on 31/Dec/22

	$w o w, t h a n k s s i r$
	Answered by manxsol last updated on 31/Dec/22
	$(√a)−(√b)=1 ((√a)+(√b))((√a)−(√b))=(√a)+(√b) (√a)+(√b)=19 add 2(√a)=20⇒ (√a)=10 (√b)=9 5x^2 −7x+10=100 5x^2 −7x−90=0 x1=5 x2=−((18)/5) 5x^2 −7x−9=81 5x^2 −7x−90=0 equal equation comprobation x1✓ x2✓ csolution={5,−((18)/5)}$
	$\sqrt{a} - \sqrt{b} = 1$ $(\sqrt{a} + \sqrt{b}) (\sqrt{a} - \sqrt{b}) = \sqrt{a} + \sqrt{b}$ $\sqrt{a} + \sqrt{b} = 19$ $a d d$ $2 \sqrt{a} = 20 \Rightarrow \sqrt{a} = 10 \sqrt{b} = 9$ $5 x^{2} - 7 x + 10 = 100$ $5 x^{2} - 7 x - 90 = 0$ $x 1 = 5$ $x 2 = - \frac{18}{5}$ $5 x^{2} - 7 x - 9 = 81$ $5 x^{2} - 7 x - 90 = 0 e q u a l e q u a t i o n$ $c o m p r o b a t i o n$ $x 1 ✓$ $x 2 ✓$ $c s o l u t i o n = {5, - \frac{18}{5}}$
	Commented by Michaelfaraday last updated on 31/Dec/22

	$t h a n k s s i r$
	Answered by Rasheed.Sindhi last updated on 31/Dec/22

	$\sqrt{5 x^{2} - 7 x + 10} - \sqrt{5 x^{2} - 7 x - 9} = 1 . . (i)$ $M u l t i p l y i n g b y \sqrt{5 x^{2} - 7 x + 10} + \sqrt{5 x^{2} - 7 x - 9} t o b o t h s i d e s :$ $(5 x^{2} - 7 x + 10) - (5 x^{2} - 7 x - 9)$ $= \sqrt{5 x^{2} - 7 x + 10} + \sqrt{5 x^{2} - 7 x - 9}$ $\sqrt{5 x^{2} - 7 x + 10} + \sqrt{5 x^{2} - 7 x - 9} = 19 . . . (i i)$ $(i) + (i i)$ $2 \sqrt{5 x^{2} - 7 x + 10} = 20$ $5 x^{2} - 7 x + 10 = 100$ $5 x^{2} - 7 x - 90 = 0$ $(x - 5) (5 x + 18) = 0$ $x = 5 \lor x = - \frac{18}{5}$
	Commented by Michaelfaraday last updated on 31/Dec/22

	$t h a n k s s i r$
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