Solve and Compare: Delving into (√x+8)(√x-8)=0 and x²-100=0

Solve each equation separately and find which x is the largest.

  1. (x+8)(x−8)=0 (\sqrt{x}+8)(\sqrt{x}-8)=0

  2. x2−100=0 x^2-100=0

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the largest X
00:03 First let's find X from section 1
00:10 Find the solutions that make the equation equal to zero
00:18 Isolate X
00:23 The root of a number will always be greater than or equal to 0
00:26 Minus 8 is less than 0, therefore there is no solution in this case
00:30 The second possible solution
00:36 Isolate X, and square both sides to eliminate the root
00:43 This is the solution for X in section 1
00:53 Now let's calculate X from section 2
01:00 Isolate X
01:11 Take the root to find the possible solutions for X
01:15 These are the possible solutions for X in section 2
01:18 We can see that in section 1 one of the solutions is the largest
01:21 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve each equation separately and find which x is the largest.

  1. (x+8)(x−8)=0 (\sqrt{x}+8)(\sqrt{x}-8)=0

  2. x2−100=0 x^2-100=0

2

Step-by-step solution

To solve each equation separately and find the largest value for x x , follow these steps:

  • Step 1: Solve (x+8)(x−8)=0 (\sqrt{x}+8)(\sqrt{x}-8)=0 .
  • Step 2: Solve x2−100=0 x^2-100=0 .
  • Step 3: Compare the solutions to determine the largest x x .

Now, let's solve each step:

**Step 1**: For the equation (x+8)(x−8)=0 (\sqrt{x}+8)(\sqrt{x}-8)=0 , apply the zero-product property:

- x+8=0\sqrt{x}+8=0 or x−8=0\sqrt{x}-8=0.
Solving these, we get:
From x+8=0\sqrt{x}+8=0:
x=−8\sqrt{x} = -8, which has no real solutions since square roots cannot be negative.
From x−8=0\sqrt{x}-8=0:
x=8\sqrt{x} = 8, thus x=82=64x = 8^2 = 64.

**Step 2**: For the equation x2−100=0 x^2-100=0 , recognize it as a difference of squares x2−102=0 x^2-10^2=0 :

- Factoring gives (x−10)(x+10)=0(x-10)(x+10)=0.
Solving these, we find:
x−10=0x-10=0, so x=10x=10.
x+10=0x+10=0, so x=−10x=-10.

**Step 3**: Compare the solutions of both equations to determine the largest x x :

- From Step 1, the valid solution is x=64x=64. - From Step 2, the valid solutions are x=10x=10 and x=−10x=-10.
The largest x x is 6464 from the solutions obtained.

Therefore, the largest solution for x x is x=64 x = 64 .

3

Final Answer

1

Practice Quiz

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\( (2+x)(2-x)=0 \)

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