Find Equivalent Expressions for (x-7)²: Perfect Square Expansion

Q: Why is the middle term negative when I'm squaring?

When you square $ (x-7) $, you're multiplying $ (x-7)(x-7) $. The middle terms are $ -7x $ and $ -7x $, which combine to give -14x. The negative comes from the original subtraction!

Q: How do I remember the perfect square formula?

Think "First, Twice, Last": First term squared, twice the product of both terms, last term squared. For $ (x-7)^2 $: x² (first), -14x (twice), 49 (last).

Q: What's the difference between (x-7)² and (x+7)²?

Only the middle term changes sign! $ (x-7)^2 = x^2 - 14x + 49 $ but $ (x+7)^2 = x^2 + 14x + 49 $. The first and last terms stay the same.

Q: Can I just distribute (x-7)(x-7) instead?

Absolutely! Using FOIL gives the same result: $ x \cdot x = x^2 $, $ x \cdot (-7) = -7x $, $ (-7) \cdot x = -7x $, $ (-7) \cdot (-7) = 49 $. Combine to get $ x^2 - 14x + 49 $.

Q: How do I check if x² - 14x + 49 factors back to (x-7)²?

Look for a perfect square trinomial pattern: $ a^2 - 2ab + b^2 $. Here, $ x^2 $ and $ 49 = 7^2 $ are perfect squares, and $ -14x = -2(x)(7) $. So it factors to $ (x-7)^2 $!

Perfect Square Binomials with Subtraction

Choose the expression that has the same value as the following:

$(x-7)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 We will use the shortened multiplication formulas to expand the brackets

00:15 X is the A in the formula

00:18 And 7 is the B in the formula

00:28 We'll substitute according to the formula and get the expansion of the brackets

00:36 We'll solve the multiplications

00:41 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that has the same value as the following:

$(x-7)^2$

Step-by-step solution

To solve the problem, we need to expand the expression $(x-7)^2$ using the formula for the square of a difference.

The formula for the square of a difference is $(a-b)^2 = a^2 - 2ab + b^2$ .

Let's apply this formula to our expression $(x-7)^2$ :

Identify $a = x$ and $b = 7$ .
Substitute these values into the formula: $(x-7)^2 = x^2 - 2(x)(7) + 7^2$ .
Calculate each term:

$x^2$ remains as $x^2$ .
$-2(x)(7) = -14x$ .
$7^2 = 49$ .

So, expanding the expression, we get $x^2 - 14x + 49$ .

Thus, the expression that has the same value as $(x-7)^2$ is $x^2 - 14x + 49$ .

Final Answer

$x^2-14x+49$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $(a-b)^2 = a^2 - 2ab + b^2$ for difference squares
Technique: For $(x-7)^2$ : middle term is $-2(x)(7) = -14x$
Check: Expand $x^2 - 14x + 49$ back to $(x-7)^2$ using factoring ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative sign in the middle term
Don't write $(x-7)^2 = x^2 + 14x + 49$ by copying the plus sign! This ignores the subtraction in the original binomial and gives a completely wrong expansion. Always remember that $(a-b)^2$ has a negative middle term: $-2ab$ .

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

$$ (7b-3x)^2 $$

FAQ

Everything you need to know about this question

Why is the middle term negative when I'm squaring?

When you square $(x-7)$ , you're multiplying $(x-7)(x-7)$ . The middle terms are $-7x$ and $-7x$ , which combine to give -14x. The negative comes from the original subtraction!

How do I remember the perfect square formula?

Think "First, Twice, Last": First term squared, twice the product of both terms, last term squared. For $(x-7)^2$ : x² (first), -14x (twice), 49 (last).

What's the difference between (x-7)² and (x+7)²?

Only the middle term changes sign! $(x-7)^2 = x^2 - 14x + 49$ but $(x+7)^2 = x^2 + 14x + 49$ . The first and last terms stay the same.

Can I just distribute (x-7)(x-7) instead?

Absolutely! Using FOIL gives the same result: $x \cdot x = x^2$ , $x \cdot (-7) = -7x$ , $(-7) \cdot x = -7x$ , $(-7) \cdot (-7) = 49$ . Combine to get $x^2 - 14x + 49$ .

How do I check if x² - 14x + 49 factors back to (x-7)²?

Look for a perfect square trinomial pattern: $a^2 - 2ab + b^2$ . Here, $x^2$ and $49 = 7^2$ are perfect squares, and $-14x = -2(x)(7)$ . So it factors to $(x-7)^2$ !

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