Solve each equation separately and find which x is the largest.$ (\sqrt{x}+8)(\sqrt{x}-8)=0 $$ x^2-100=0 $

There is no solution to the two equations

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

Q: Why can't √x equal a negative number like -8?

By definition, the principal square root of any real number is always non-negative. $ \sqrt{x} = -8 $ has no real solution because square roots cannot be negative.

Q: How do I solve equations with square roots?

First, isolate the square root on one side. Then square both sides to eliminate the radical. Always check your answers in the original equation!

Q: What's the difference between x² - 100 = 0 and the radical equation?

The equation $ x^2 - 100 = 0 $ gives two solutions: x = 10 and x = -10. The radical equation has domain restrictions and only gives x = 64.

Q: Why is 64 larger than both 10 and -10?

When comparing numbers, 64 > 10 > -10. The question asks for the largest x-value from all solutions combined, which is 64 from the first equation.

Q: Do I need to check both equations separately?

Yes! Solve each equation independently first, then compare all solutions to find the largest value. Don't mix the solutions during solving.

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.

$(\sqrt{x}+8)(\sqrt{x}-8)=0$
$x^2-100=0$

2

Step-by-step solution

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

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

Now, let's solve each step:

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

-

\sqrt{x}+8=0

or

\sqrt{x}-8=0

.
Solving these, we get:
From

\sqrt{x}+8=0

:

\sqrt{x} = -8

, which has no real solutions since square roots cannot be negative.
From

\sqrt{x}-8=0

:

\sqrt{x} = 8

, thus

x = 8^2 = 64

.

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

- Factoring gives

(x-10)(x+10)=0

.
Solving these, we find:

x-10=0

, so

x=10

.

x+10=0

, so

x=-10

.

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

- From Step 1, the valid solution is

x=64

. - From Step 2, the valid solutions are

x=10

and

x=-10

.
The largest

x

is

64

from the solutions obtained.

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

3

Final Answer

1

Key Points to Remember

Essential concepts to master this topic

Domain: Square roots require non-negative values under the radical
Technique: Apply zero-product property: $(\sqrt{x}+8)(\sqrt{x}-8)=0$ gives $\sqrt{x}=8$
Check: Verify $x=64$ : $(\sqrt{64}+8)(\sqrt{64}-8) = 16 \cdot 0 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Accepting negative values for square roots
Don't solve $\sqrt{x} = -8$ and get $x = 64$ = impossible solution! Square roots of real numbers are always non-negative by definition. Always reject solutions where $\sqrt{x}$ equals a negative number.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why can't √x equal a negative number like -8?

+

By definition, the principal square root of any real number is always non-negative. $\sqrt{x} = -8$ has no real solution because square roots cannot be negative.

How do I solve equations with square roots?

+

First, isolate the square root on one side. Then square both sides to eliminate the radical. Always check your answers in the original equation!

What's the difference between x² - 100 = 0 and the radical equation?

+

The equation $x^2 - 100 = 0$ gives two solutions: x = 10 and x = -10. The radical equation has domain restrictions and only gives x = 64.

Why is 64 larger than both 10 and -10?

+

When comparing numbers, 64 > 10 > -10. The question asks for the largest x-value from all solutions combined, which is 64 from the first equation.

Do I need to check both equations separately?

+

Yes! Solve each equation independently first, then compare all solutions to find the largest value. Don't mix the solutions during solving.

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

Radical Equations with Domain Restrictions

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