Express \[$(x+12)^2\$\] As A Trinomial In Standard Form.

What is a Perfect Square Trinomial?

A perfect square trinomial is a trinomial that can be factored into the square of a binomial. It has the form $(a+b)^2$ or $(a-b)^2$, where $a$ and $b$ are constants or variables. In this case, we are given the expression $(x+12)^2$ and we need to expand it into a trinomial in standard form.

Expanding the Perfect Square Trinomial

To expand the perfect square trinomial, we can use the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, $a = x$ and $b = 12$. Plugging these values into the formula, we get:

$(x+12)^2 = x^2 + 2(x)(12) + 12^2$

Simplifying the Expression

Now, we can simplify the expression by evaluating the products and adding the terms.

$x^2 + 2(x)(12) + 12^2 = x^2 + 24x + 144$

Standard Form of a Trinomial

A trinomial is in standard form when it is written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, the trinomial is already in standard form:

$x^2 + 24x + 144$

Conclusion

In this article, we have expanded the perfect square trinomial $(x+12)^2$ into a trinomial in standard form. We have used the formula $(a+b)^2 = a^2 + 2ab + b^2$ to expand the trinomial and simplified the expression by evaluating the products and adding the terms. The final answer is $x^2 + 24x + 144$.

Example Problems

Example 1: Expanding a Perfect Square Trinomial

Expand the perfect square trinomial $(x+5)^2$ into a trinomial in standard form.

Solution

Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, we get:

$(x+5)^2 = x^2 + 2(x)(5) + 5^2$ $= x^2 + 10x + 25$

Example 2: Expanding a Perfect Square Trinomial

Expand the perfect square trinomial $(x-3)^2$ into a trinomial in standard form.

Solution

Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, we get:

$(x-3)^2 = x^2 + 2(x)(-3) + (-3)^2$ $= x^2 - 6x + 9$

Tips and Tricks

• When expanding a perfect square trinomial, use the formula $(a+b)^2 = a^2 + 2ab + b^2$.
• Simplify the expression by evaluating the products and adding the terms.
• Check that the trinomial is in standard form by writing it in the form $ax^2 + bx + c$.

Practice Problems

Problem 1: Expanding a Perfect Square Trinomial

Expand the perfect square trinomial $(x+8)^2$ into a trinomial in standard form.

Problem 2: Expanding a Perfect Square Trinomial

Expand the perfect square trinomial $(x-2)^2$ into a trinomial in standard form.

Answer Key

Problem 1: Expanding a Perfect Square Trinomial

$x^2 + 16x + 64$

Problem 2: Expanding a Perfect Square Trinomial

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Frequently Asked Questions

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be factored into the square of a binomial. It has the form $(a+b)^2$ or $(a-b)^2$, where $a$ and $b$ are constants or variables.

Q: How do I expand a perfect square trinomial?

A: To expand a perfect square trinomial, use the formula $(a+b)^2 = a^2 + 2ab + b^2$. Plug in the values of $a$ and $b$ and simplify the expression.

Q: What is the standard form of a trinomial?

A: A trinomial is in standard form when it is written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I simplify a perfect square trinomial?

A: To simplify a perfect square trinomial, evaluate the products and add the terms. For example, if you have $(x+12)^2$, you would evaluate $x^2 + 2(x)(12) + 12^2$ and simplify to get $x^2 + 24x + 144$.

Q: Can I use the formula $(a+b)^2 = a^2 + 2ab + b^2$ to expand any trinomial?

A: No, the formula $(a+b)^2 = a^2 + 2ab + b^2$ is only used to expand perfect square trinomials. If you have a trinomial that is not a perfect square, you will need to use a different method to expand it.

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

A: To determine if a trinomial is a perfect square, look for the form $(a+b)^2$ or $(a-b)^2$. If the trinomial can be written in one of these forms, it is a perfect square.

Q: Can I use the formula $(a+b)^2 = a^2 + 2ab + b^2$ to factor a trinomial?

A: No, the formula $(a+b)^2 = a^2 + 2ab + b^2$ is used to expand perfect square trinomials, not to factor them. If you need to factor a trinomial, you will need to use a different method.

Common Mistakes

Mistake 1: Not using the correct formula

• Make sure to use the formula $(a+b)^2 = a^2 + 2ab + b^2$ to expand perfect square trinomials.
• Do not use this formula to factor trinomials.

Mistake 2: Not simplifying the expression

• Make sure to evaluate the products and add the terms when simplifying a perfect square trinomial.
• Do not leave the expression in its expanded form.

Mistake 3: Not checking the standard form

• Make sure to write the trinomial in standard form ($ax^2 + bx + c$) after expanding it.
• Do not leave the trinomial in its expanded form.

Tips and Tricks

• When expanding a perfect square trinomial, use the formula $(a+b)^2 = a^2 + 2ab + b^2$.
• Simplify the expression by evaluating the products and adding the terms.
• Check that the trinomial is in standard form by writing it in the form $ax^2 + bx + c$.

Practice Problems

Problem 1: Expanding a Perfect Square Trinomial

Expand the perfect square trinomial $(x+9)^2$ into a trinomial in standard form.

Problem 2: Expanding a Perfect Square Trinomial

Expand the perfect square trinomial $(x-6)^2$ into a trinomial in standard form.

Answer Key

Problem 1: Expanding a Perfect Square Trinomial

$x^2 + 18x + 81$

Problem 2: Expanding a Perfect Square Trinomial

$x^2 - 12x + 36$