Problem of the Week

Updated at Mar 4, 2019 1:58 PM

This week we have another equation problem:

How would you solve the equation 6(52+w)2=2566{(\frac{5}{2+w})}^{2}=\frac{25}{6}?

Let's start!



6(52+w)2=2566{(\frac{5}{2+w})}^{2}=\frac{25}{6}

1
Use Division Distributive Property: (xy)a=xaya{(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}.
6×52(2+w)2=2566\times \frac{{5}^{2}}{{(2+w)}^{2}}=\frac{25}{6}

2
Simplify  52{5}^{2}  to  2525.
6×25(2+w)2=2566\times \frac{25}{{(2+w)}^{2}}=\frac{25}{6}

3
Simplify  6×25(2+w)26\times \frac{25}{{(2+w)}^{2}}  to  150(2+w)2\frac{150}{{(2+w)}^{2}}.
150(2+w)2=256\frac{150}{{(2+w)}^{2}}=\frac{25}{6}

4
Multiply both sides by (2+w)2{(2+w)}^{2}.
150=256(2+w)2150=\frac{25}{6}{(2+w)}^{2}

5
Simplify  256(2+w)2\frac{25}{6}{(2+w)}^{2}  to  25(2+w)26\frac{25{(2+w)}^{2}}{6}.
150=25(2+w)26150=\frac{25{(2+w)}^{2}}{6}

6
Multiply both sides by 66.
150×6=25(2+w)2150\times 6=25{(2+w)}^{2}

7
Simplify  150×6150\times 6  to  900900.
900=25(2+w)2900=25{(2+w)}^{2}

8
Divide both sides by 2525.
90025=(2+w)2\frac{900}{25}={(2+w)}^{2}

9
Simplify  90025\frac{900}{25}  to  3636.
36=(2+w)236={(2+w)}^{2}

10
Take the square root of both sides.
±36=2+w\pm \sqrt{36}=2+w

11
Since 6×6=366\times 6=36, the square root of 3636 is 66.
±6=2+w\pm 6=2+w

12
Switch sides.
2+w=±62+w=\pm 6

13
Break down the problem into these 2 equations.
2+w=62+w=6
2+w=−62+w=-6

14
Solve the 1st equation: 2+w=62+w=6.
w=4w=4

15
Solve the 2nd equation: 2+w=−62+w=-6.
w=−8w=-8

16
Collect all solutions.
w=4,−8w=4,-8

Done
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