Problem of the Week

Updated at Mar 23, 2020 11:51 AM

For this week we've brought you this equation problem.

How can we solve the equation (4z−4)2−6=58{(4z-4)}^{2}-6=58?

Here are the steps:



(4z−4)2−6=58{(4z-4)}^{2}-6=58

1
Factor out the common term 44.
(4(z−1))2−6=58{(4(z-1))}^{2}-6=58

2
Use Multiplication Distributive Property: (xy)a=xaya{(xy)}^{a}={x}^{a}{y}^{a}.
42(z−1)2−6=58{4}^{2}{(z-1)}^{2}-6=58

3
Simplify  42{4}^{2}  to  1616.
16(z−1)2−6=5816{(z-1)}^{2}-6=58

4
Add 66 to both sides.
16(z−1)2=58+616{(z-1)}^{2}=58+6

5
Simplify  58+658+6  to  6464.
16(z−1)2=6416{(z-1)}^{2}=64

6
Divide both sides by 1616.
(z−1)2=6416{(z-1)}^{2}=\frac{64}{16}

7
Simplify  6416\frac{64}{16}  to  44.
(z−1)2=4{(z-1)}^{2}=4

8
Take the square root of both sides.
z−1=±4z-1=\pm \sqrt{4}

9
Since 2×2=42\times 2=4, the square root of 44 is 22.
z−1=±2z-1=\pm 2

10
Break down the problem into these 2 equations.
z−1=2z-1=2
z−1=−2z-1=-2

11
Solve the 1st equation: z−1=2z-1=2.
z=3z=3

12
Solve the 2nd equation: z−1=−2z-1=-2.
z=−1z=-1

13
Collect all solutions.
z=3,−1z=3,-1

Done
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