101st Problem 2016

Algebra Level 2

Solve the quadratic equation:

25 x 2 + 10 x = − 1 25x^2 + 10x = -1

Check out the set: 2016 Problems .
x = ± 1 5 \large x=\pm \dfrac { 1 }{ 5 } x = − 5 \large x=-5 x = − 1 5 only \large x=- \dfrac { 1 }{ 5 } \text{ only} x = ± 5 only \large x=\pm 5 \text{ only}

Angela Fajardo by Angela Fajardo , 19, Philippines

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2 solutions

Ashish Menon
Apr 22, 2016

25 x 2 + 10 x = − 1 25 x 2 + 10 x + 1 = 0 25 x 2 + 5 x + 5 x + 1 = 0 5 x ( 5 x + 1 ) + 1 ( 5 x + 1 ) = 0 ( 5 x + 1 ) 2 = 0 5 x + 1 = 0 x = − 1 5 \begin{aligned} 25x^2 + 10x & = -1\\ 25x^2 + 10x + 1 & = 0\\ 25x^2 + 5x + 5x + 1 & = 0\\ 5x(5x + 1) + 1(5x + 1) & = 0\\ {(5x + 1)}^2 & = 0\\ 5x + 1 & = 0\\ x & = \boxed{-\dfrac{1}{5}} \end{aligned}

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Pham Khanh
Apr 24, 2016

25 x 2 + 10 x = − 1 25x^{2}+10x=-1 ⟺ ( 5 x ) 2 + 10 x + 1 = 0 \iff (5x)^{2}+10x+1=0 ⟺ ( 5 x ) 2 + 5 x × 2 × 1 + 1 2 = 0 \iff (5x)^{2}+5x \times 2 \times 1+1^{2}=0 ⟺ ( 5 x + 1 ) 2 = 0 \iff (5x+1)^{2}=0 ⟺ 5 x + 1 = ± 0 = 0 \iff 5x+1= \pm 0=0 ⟺ x = 0 − 1 5 = − 1 5 o n l y \iff x=\frac{0-1}{5}=\Large \boxed{\frac{-1}{5} only}

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Moderator note:

Simple standard approach.

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