Squares won't help you... again.

Algebra Level 3

Solve the following equation:

2 x + 8 + x 2 + 5 x + 6 = 5 x 2 + 50 x + 105 \large 2\sqrt{x + 8} + \sqrt{x^2 + 5x + 6} = \sqrt{5x^2 + 50x + 105}

If x = a + b c x = \dfrac{\sqrt a + b}{c} then calculate a + b c \dfrac{a + b}{c} .


The answer is 63.25.

Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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1 solution

2 x + 8 + x 2 + 5 x + 6 = 5 x 2 + 50 x + 105 . S q u a r i n g b o t h s i d e s , 4 x + 32 + x 2 + 5 x + 6 + 4 ( x + 8 ) ( x 2 + 5 x + 6 ) = 5 x 2 + 50 x + 105. ∴ 4 ( x + 8 ) ( x 2 + 5 x + 6 ) = 4 x 2 + 41 x + 67. A g a i n s q u a r i n g b o t h s i d e s , 16 ( x + 8 ) ( x 2 + 5 x + 6 ) = ( 4 x 2 + 41 x + 67 ) 2 . ∴ 16 x 4 + 312 x 3 + 2009 x 2 + 4758 x + 3721 = 0. ⟹ ( 4 x 2 + 39 x + 61 ) 2 = 0. S o l v i n g q u a d r a t i c , x = − 39 ± 3 9 2 − 4 ∗ 4 ∗ 61 2 ∗ 4 . x = − 39 ± 545 8 . x = b ± a c . ± a + b c = ± 545 − 39 8 = 63.25 O R − 73 2\sqrt{x+8}+\sqrt{x^2+5x+6}=\sqrt{5x^2+50x+105}.\\ ~~~~\\ Squaring~both~sides,\\ 4x+32+x^2+5x+6+4\sqrt{(x+8)(x^2+5x+6)}=5x^2+50x+105.\\ \therefore~~4\sqrt{(x+8)(x^2+5x+6)}=4x^2+41x+67.\\ ~~~\\ Again~squaring~both~sides,\\ 16(x+8)(x^2+5x+6)=(4x^2+41x+67)^2.\\ \therefore~~16x^4+312x^3+2009x^2+4758x+3721=0.\\ \implies~(4x^2+39x+61)^2=0.\\ ~~~\\ Solving~quadratic,\\ x=\dfrac{-39\pm\sqrt{39^2-4*4*61} }{2*4}.\\ x=\dfrac{-39\pm\sqrt{545} }{8}.\\ x=\dfrac{b\pm\sqrt{a} }{c}.\\ ~~~\\ \dfrac{\pm a+b} c=\dfrac{\pm 545-39} 8=\color{#D61F06}{\Large~63.25~~OR~~-73}\\
Why the heading "Squares won't help you... again." !!!

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