Solve (x-√7)(x+√7) = x²+7x+7: Square Root Equation Challenge

Q: How do I recognize when to use the difference of squares formula?

Look for the pattern $ (a-b)(a+b) $ where you have the same terms but with opposite signs. Here, we have $ (x-\sqrt{7})(x+\sqrt{7}) $ which fits perfectly!

Q: What happens to the middle terms when I expand?

In difference of squares, the middle terms always cancel out! $ x\sqrt{7} - x\sqrt{7} = 0 $, leaving only $ x^2 - 7 $.

Q: Why did the equation become linear instead of quadratic?

After simplifying, we got $ x^2 - 7 = x^2 + 7x + 7 $. When we subtract $ x^2 $ from both sides, the quadratic terms cancel, leaving us with the linear equation $ -7 = 7x + 7 $.

Q: How can I check if x = -2 is correct?

Substitute into the original equation:Left side: $ (-2-\sqrt{7})(-2+\sqrt{7}) = (-2)^2 - (\sqrt{7})^2 = 4 - 7 = -3 $Right side: $ (-2)^2 + 7(-2) + 7 = 4 - 14 + 7 = -3 $Both sides equal -3, so our answer is correct! ✓

Q: What if I don't remember the difference of squares formula?

You can still solve by expanding, but it takes longer! $ (x-\sqrt{7})(x+\sqrt{7}) = x^2 + x\sqrt{7} - x\sqrt{7} - 7 = x^2 - 7 $. The middle terms will always cancel, giving the same result.

Difference of Squares with Linear Solutions

Solve the following equation:

_{$(x-\sqrt{7})(x+\sqrt{7})=x^2+7x+7$}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's solve a math problem together.

00:16 We'll use some handy shortcuts for multiplication. Ready?

00:36 First, let's calculate the square root of 7.

00:44 Now, let's simplify what we can. Step by step.

00:50 Next, we need to isolate X. Almost there!

01:07 And that's how we find the solution to our problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

_{$(x-\sqrt{7})(x+\sqrt{7})=x^2+7x+7$}

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Simplify the equation using the difference of squares.
Step 2: Rearrange to form a suitable quadratic equation.
Step 3: Solve the quadratic equation to find the value of $x$ .

Now, let's work through each step:

Step 1: Simplify the left side using the difference of squares formula:

$(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7$

Step 2: Set the expressions equal and form a quadratic:

$x^2 - 7 = x^2 + 7x + 7$

Subtract $x^2$ from both sides:

$-7 = 7x + 7$

Rearrange the equation to isolate $x$ :

$7x + 7 = -7$

Subtract 7 from both sides:

$7x = -14$

Divide both sides by 7:

$x = -2$

Therefore, the solution to the equation is $x = -2$ .

Final Answer

-2

Key Points to Remember

Essential concepts to master this topic

Formula: $(a-b)(a+b) = a^2 - b^2$ simplifies the left side
Technique: $(x-\sqrt{7})(x+\sqrt{7}) = x^2 - 7$ by difference of squares
Check: Substitute x = -2: $(-2)^2 - 7 = 4 - 7 = -3$ matches right side ✓

Common Mistakes

Avoid these frequent errors

Expanding the left side by distribution instead of using difference of squares
Don't expand $(x-\sqrt{7})(x+\sqrt{7})$ as $x^2 + x\sqrt{7} - x\sqrt{7} - 7$ = more work and chance for errors! This makes the problem unnecessarily complicated. Always recognize the difference of squares pattern and use $a^2 - b^2$ directly.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

How do I recognize when to use the difference of squares formula?

Look for the pattern $(a-b)(a+b)$ where you have the same terms but with opposite signs. Here, we have $(x-\sqrt{7})(x+\sqrt{7})$ which fits perfectly!

What happens to the middle terms when I expand?

In difference of squares, the middle terms always cancel out! $x\sqrt{7} - x\sqrt{7} = 0$ , leaving only $x^2 - 7$ .

Why did the equation become linear instead of quadratic?

After simplifying, we got $x^2 - 7 = x^2 + 7x + 7$ . When we subtract $x^2$ from both sides, the quadratic terms cancel, leaving us with the linear equation $-7 = 7x + 7$ .

How can I check if x = -2 is correct?

Substitute into the original equation:

Left side: $(-2-\sqrt{7})(-2+\sqrt{7}) = (-2)^2 - (\sqrt{7})^2 = 4 - 7 = -3$
Right side: $(-2)^2 + 7(-2) + 7 = 4 - 14 + 7 = -3$
Both sides equal -3, so our answer is correct! ✓

What if I don't remember the difference of squares formula?

You can still solve by expanding, but it takes longer! $(x-\sqrt{7})(x+\sqrt{7}) = x^2 + x\sqrt{7} - x\sqrt{7} - 7 = x^2 - 7$ . The middle terms will always cancel, giving the same result.

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