100th Problem 2016

Algebra Level 1

Solve the quadratic equation

( 5 x − 3 ) 2 = 32. (5x-3)^2 = 32 .

Check out the set: 2016 Problems .
x = 3 ± 4 2 5 \large x= \dfrac { 3 \pm 4\sqrt { 2 } }{ 5 } x = ± 3.8 \large x=\pm 3.8 x = ± 7 \large x=\pm 7

Angela Fajardo by Angela Fajardo , 19, Philippines

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3 solutions

Ashish Menon
Apr 22, 2016

( 5 x − 3 ) 2 = 32 25 x 2 − 30 x + 9 = 32 25 x 2 − 30 x − 23 = 0 Using quadratic formula a = 25 ; b = 30 ; c = − 23 x = − b ± b 2 − 4 a c 2 a x = − ( − 30 ) ± ( − 30 ) 2 − 4 × 25 × − 23 2 × 25 x = 30 ± 900 + 2300 50 x = 30 ± 3200 50 x = 30 ± 1600 × 2 50 x = 30 ± 40 2 50 x = 10 ( 3 ± 4 2 ) 50 x = 3 ± 4 2 5 \begin{aligned} {(5x - 3)}^2 & = 32\\ 25x^2 - 30x + 9 & =32\\ 25x^2 - 30x - 23 & = 0\\ \text{Using quadratic formula}\\ a=25 ; b=30 ; c = -23\\ x & = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\ \\ x & = \dfrac{-(-30) \pm \sqrt{{(-30)}^2 - 4×25×-23}}{2×25}\\ \\ x & = \dfrac{30 \pm \sqrt{900 + 2300}}{50}\\ \\ x & = \dfrac{30 \pm \sqrt{3200}}{50}\\ \\ x & = \dfrac{30 \pm \sqrt{1600 × 2}}{50}\\ \\ x & = \dfrac{30 \pm 40\sqrt{2}}{50}\\ \\ x & = \dfrac{10(3 \pm 4\sqrt{2})}{50}\\ \\ x & = \boxed{\dfrac{3 \pm 4\sqrt{2}}{5}} \end{aligned}

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The second last row has a slight error in the display

Hung Woei Neoh - 5 years, 1 month ago

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Thanks, I corrected it :)

Ashish Menon - 5 years, 1 month ago

Well, you could just take the square root to each side.

. . - 1 month, 1 week ago
. .
May 8, 2021

( 5 x − 3 ) 2 = 32 ( 5x - 3 ) ^ { 2 } = 32

5 x − 3 = ± 4 2 5x - 3 = \pm 4 \sqrt { 2 }

5 x = 3 ± 4 2 5x = 3 \pm 4 \sqrt { 2 }

x = 3 ± 4 2 5 x = \frac { 3 \pm 4 \sqrt { 2 } } { 5 }

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Neel Aryan
Apr 22, 2016

Shouldn't the plus minus sign be in front of 4√2 rather than the whole fraction?

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Indeed, a report has pointed that out already. Error in my part; thanks anyway :)

Angela Fajardo - 5 years, 1 month ago

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