Solve x²-2x+1=9: Using the Abbreviated Multiplication Formula

Q: How do I recognize a perfect square trinomial?

Look for the pattern $ a^2 - 2ab + b^2 $ or $ a^2 + 2ab + b^2 $. In $ x^2 - 2x + 1 $, we have first term squared (x²), twice the product (-2x), and second term squared (1²).

Q: Why can't I just expand and use the quadratic formula?

You absolutely can! But recognizing the perfect square is much faster. The abbreviated multiplication formula saves you from messy calculations and gives you the answer in fewer steps.

Perfect Square Trinomials with Square Root Method

$x^2-2x+1=9$

Solve using the abbreviated multiplication formula

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve using the shortened multiplication formula

00:03 Use shortened multiplication formulas to find the parentheses

00:13 In this case X is A

00:19 and 1 is B

00:33 From this we'll find the parentheses

00:38 Extract the root

00:47 Find the answers for 2 options (positive and negative)

00:51 Isolate X

00:55 This is one solution

01:02 Isolate X in the second solution

01:04 Isolate X

01:08 This is the second solution

01:11 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

$x^2-2x+1=9$

Solve using the abbreviated multiplication formula

Step-by-step solution

We will solve the quadratic equation $x^2 - 2x + 1 = 9$ using the square of a binomial formula.

Firstly, let's recognize that the left side of the equation forms a perfect square:

$x^2 - 2x + 1 \equiv (x - 1)^2$

Therefore, the equation can be rewritten as:

$(x - 1)^2 = 9$

To solve for $x$ , take the square root of both sides. Remember to consider both the positive and negative solutions from the square root:

Thus, $x - 1 = \pm 3$

This gives us two separate equations to solve:

$x - 1 = 3$
$x - 1 = -3$

Solving each equation for $x$ gives:

For $x - 1 = 3$ :

Add 1 to both sides: $x = 4$

For $x - 1 = -3$ :

Add 1 to both sides: $x = -2$

Therefore, the solutions to the equation are $x = 4$ and $x = -2$ .

Comparing these solutions to the given answer choices, we identify the correct choice as:

$x=-2$ or $x=4$

In conclusion, the solutions to the equation are $x = 4$ and $x = -2$ .

Final Answer

$x=-2$ o $x=4$

Key Points to Remember

Essential concepts to master this topic

Recognition: Identify $x^2 - 2x + 1$ as $(x-1)^2$ perfect square
Technique: Rewrite equation as $(x-1)^2 = 9$ , then $x-1 = ±3$
Check: Substitute $x = 4$ : $16 - 8 + 1 = 9$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the ± when taking square roots
Don't write x - 1 = 3 only = missing half the solutions! Taking the square root of 9 gives both +3 and -3, so you get two different values for x. Always write x - 1 = ±3 to find both solutions x = 4 and x = -2.

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

How do I recognize a perfect square trinomial?

Look for the pattern $a^2 - 2ab + b^2$ or $a^2 + 2ab + b^2$ . In $x^2 - 2x + 1$ , we have first term squared (x²), twice the product (-2x), and second term squared (1²).

Why can't I just expand and use the quadratic formula?

You absolutely can! But recognizing the perfect square is much faster. The abbreviated multiplication formula saves you from messy calculations and gives you the answer in fewer steps.

What if the right side wasn't a perfect square like 9?

No problem! You'd still take the square root of both sides. For example, if $(x-1)^2 = 7$ , then $x-1 = ±\sqrt{7}$ , giving $x = 1 ± \sqrt{7}$ .

How do I know which answer choice is correct?

Calculate both solutions and match them to the choices. Here we got x = 4 and x = -2, which matches the fourth option. Always solve completely before looking at the choices!

Can I verify my answers?

Yes! Substitute each solution back into the original equation. For x = 4: $4^2 - 2(4) + 1 = 16 - 8 + 1 = 9$ ✓. For x = -2: $(-2)^2 - 2(-2) + 1 = 4 + 4 + 1 = 9$ ✓.

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