Simplify the Quadratic Expression: 4x²+20x+25

Q: How do I know if this is a perfect square trinomial?

Check if the first and last terms are perfect squares, and the middle term equals $ 2ab $. Here: $ 4x^2=(2x)^2 $, $ 25=5^2 $, and $ 20x=2(2x)(5) $.

Q: What if I get the wrong answer choice?

Always verify by expanding your factored form! If $ (4x+10)^2 $ gives you $ 16x^2+80x+100 $, that's not our original expression.

Q: Can I factor this using other methods?

Yes, but recognizing the perfect square pattern is fastest! You could use grouping or the AC method, but they take longer for expressions that are perfect squares.

Q: Why is the middle term 20x and not something else?

The middle term must equal $ 2ab $ where $ a=2x $ and $ b=5 $. So $ 2(2x)(5)=20x $. This confirms it's a perfect square!

Q: What if the expression can't be factored as a perfect square?

Then you'd need other factoring methods like grouping or the quadratic formula. Perfect square trinomials are special cases that factor beautifully!

Perfect Square Trinomials with Factoring

$4x^2+20x+25=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Write the expression as a power

00:03 Factor 2X and 2X

00:12 Factor 2, 10 and X

00:17 Factor 5 and 5

00:21 Each factor multiplied by itself is actually squared

00:33 We'll use the short multiplication formulas to find the parentheses

00:56 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

$4x^2+20x+25=$

Step-by-step solution

In this task, we are asked to simplify the formula using the abbreviated multiplication formulas.

Let's take a look at the formulas:

$(x-y)^2=x^2-2xy+y^2$

$(x+y)^2=x^2+2xy+y^2$

$(x+y)\times(x-y)=x^2-y^2$

Taking into account that in the given exercise there is only addition operation, the appropriate formula is the second one:

Now let us consider, what number when multiplied by itself will equal 4 and what number when multiplied by itself will equal 25?

The answers are respectively 2 and 5:

We insert these into the formula:

$(2x+5)^2=$

$(2x+5)(2x+5)=$

$2x\times2x+2x\times5+2x\times5+5\times5=$

$4x^2+20x+25$

That means our solution is correct.

Final Answer

$(2x+5)^2$

Key Points to Remember

Essential concepts to master this topic

Pattern: Recognize form $a^2+2ab+b^2=(a+b)^2$ for perfect squares
Method: Find square roots: $\sqrt{4x^2}=2x$ and $\sqrt{25}=5$
Verify: Expand $(2x+5)^2=4x^2+20x+25$ matches original ✓

Common Mistakes

Avoid these frequent errors

Taking wrong square roots of coefficients
Don't take √4 = 2 for the coefficient of x² without considering the variable = wrong factorization! The square root of 4x² is 2x, not 2. Always include the variable when finding square roots of terms.

Practice Quiz

Test your knowledge with interactive questions

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

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

How do I know if this is a perfect square trinomial?

Check if the first and last terms are perfect squares, and the middle term equals $2ab$ . Here: $4x^2=(2x)^2$ , $25=5^2$ , and $20x=2(2x)(5)$ .

What if I get the wrong answer choice?

Always verify by expanding your factored form! If $(4x+10)^2$ gives you $16x^2+80x+100$ , that's not our original expression.

Can I factor this using other methods?

Yes, but recognizing the perfect square pattern is fastest! You could use grouping or the AC method, but they take longer for expressions that are perfect squares.

Why is the middle term 20x and not something else?

The middle term must equal $2ab$ where $a=2x$ and $b=5$ . So $2(2x)(5)=20x$ . This confirms it's a perfect square!

What if the expression can't be factored as a perfect square?

Then you'd need other factoring methods like grouping or the quadratic formula. Perfect square trinomials are special cases that factor beautifully!

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