Find The Number To Add To $x^2 - 12x$ To Make It A Perfect Square Trinomial. Write That Trinomial As The Square Of A Binomial.A. Add 12; $(x - 6)^2$ B. Add 144; \$(x - 12)^2$[/tex\] C. Add $36$; $(x

What is a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression that can be factored as the square of a binomial. It has the form $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant. Perfect square trinomials are important in algebra because they can be used to solve quadratic equations and to simplify expressions.

The Formula for a Perfect Square Trinomial

The formula for a perfect square trinomial is $(x + a)^2 = x^2 + 2ax + a^2$ or $(x - a)^2 = x^2 - 2ax + a^2$. To find the missing number to make a trinomial a perfect square, we need to identify the value of $a$.

Finding the Missing Number

Let's consider the trinomial $x^2 - 12x$. We want to find the number to add to make it a perfect square trinomial. To do this, we need to identify the value of $a$ in the formula $(x - a)^2 = x^2 - 2ax + a^2$.

Option A: Add 12

If we add 12 to the trinomial $x^2 - 12x$, we get $x^2 - 12x + 12$. This can be factored as $(x - 6)^2$. However, this is not the correct answer because the original trinomial is $x^2 - 12x$, not $x^2 - 12x + 12$.

Option B: Add 144

If we add 144 to the trinomial $x^2 - 12x$, we get $x^2 - 12x + 144$. This can be factored as $(x - 12)^2$. However, this is not the correct answer because the original trinomial is $x^2 - 12x$, not $x^2 - 12x + 144$.

Option C: Add 36

If we add 36 to the trinomial $x^2 - 12x$, we get $x^2 - 12x + 36$. This can be factored as $(x - 6)^2$. However, this is not the correct answer because the original trinomial is $x^2 - 12x$, not $x^2 - 12x + 36$.

The Correct Answer

To find the correct answer, we need to use the formula $(x - a)^2 = x^2 - 2ax + a^2$. We know that the trinomial is $x^2 - 12x$, so we can set up the equation $x^2 - 12x = x^2 - 2ax + a^2$. Solving for $a$, we get $a = 6$. Therefore, the correct answer is to add 36 to the trinomial $x^2 - 12x$ to make it a perfect square trinomial.

Conclusion

In conclusion, a perfect square trinomial is a quadratic expression that can be factored as the square of a binomial. To find the missing number to make a trinomial a perfect square, we need to identify the value of $a$ in the formula $(x - a)^2 = x^2 - 2ax + a^2$. In this case, the correct answer is to add 36 to the trinomial $x^2 - 12x$ to make it a perfect square trinomial.

Example Problems

1. Find the missing number to make the trinomial $x^2 + 14x$ a perfect square trinomial.
2. Find the missing number to make the trinomial $x^2 - 20x$ a perfect square trinomial.
3. Find the missing number to make the trinomial $x^2 + 18x$ a perfect square trinomial.

Solutions

1. To find the missing number, we need to use the formula $(x + a)^2 = x^2 + 2ax + a^2$. We know that the trinomial is $x^2 + 14x$, so we can set up the equation $x^2 + 14x = x^2 + 2ax + a^2$. Solving for $a$, we get $a = 7$. Therefore, the missing number is 49.
2. To find the missing number, we need to use the formula $(x - a)^2 = x^2 - 2ax + a^2$. We know that the trinomial is $x^2 - 20x$, so we can set up the equation $x^2 - 20x = x^2 - 2ax + a^2$. Solving for $a$, we get $a = 10$. Therefore, the missing number is 100.
3. To find the missing number, we need to use the formula $(x + a)^2 = x^2 + 2ax + a^2$. We know that the trinomial is $x^2 + 18x$, so we can set up the equation $x^2 + 18x = x^2 + 2ax + a^2$. Solving for $a$, we get $a = 9$. Therefore, the missing number is 81.

Tips and Tricks

1. To find the missing number, use the formula $(x - a)^2 = x^2 - 2ax + a^2$ or $(x + a)^2 = x^2 + 2ax + a^2$.
2. Identify the value of $a$ in the formula.
3. Solve for $a$ using the given trinomial.
4. The missing number is the value of $a^2$.

Conclusion

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Q: What is a perfect square trinomial?

A: A perfect square trinomial is a quadratic expression that can be factored as the square of a binomial. It has the form $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant.

Q: How do I find the missing number to make a trinomial a perfect square?

A: To find the missing number, use the formula $(x - a)^2 = x^2 - 2ax + a^2$ or $(x + a)^2 = x^2 + 2ax + a^2$. Identify the value of $a$ in the formula and solve for $a$ using the given trinomial.

Q: What is the formula for a perfect square trinomial?

A: The formula for a perfect square trinomial is $(x + a)^2 = x^2 + 2ax + a^2$ or $(x - a)^2 = x^2 - 2ax + a^2$.

Q: How do I identify the value of $a$ in the formula?

A: To identify the value of $a$, look at the given trinomial and compare it to the formula. The value of $a$ is the number that is being multiplied by $x$ in the trinomial.

Q: What is the missing number in the trinomial $x^2 - 12x$?

A: To find the missing number, use the formula $(x - a)^2 = x^2 - 2ax + a^2$. We know that the trinomial is $x^2 - 12x$, so we can set up the equation $x^2 - 12x = x^2 - 2ax + a^2$. Solving for $a$, we get $a = 6$. Therefore, the missing number is 36.

Q: What is the missing number in the trinomial $x^2 + 14x$?

A: To find the missing number, use the formula $(x + a)^2 = x^2 + 2ax + a^2$. We know that the trinomial is $x^2 + 14x$, so we can set up the equation $x^2 + 14x = x^2 + 2ax + a^2$. Solving for $a$, we get $a = 7$. Therefore, the missing number is 49.

Q: What is the missing number in the trinomial $x^2 - 20x$?

A: To find the missing number, use the formula $(x - a)^2 = x^2 - 2ax + a^2$. We know that the trinomial is $x^2 - 20x$, so we can set up the equation $x^2 - 20x = x^2 - 2ax + a^2$. Solving for $a$, we get $a = 10$. Therefore, the missing number is 100.

Q: What is the missing number in the trinomial $x^2 + 18x$?

A: To find the missing number, use the formula $(x + a)^2 = x^2 + 2ax + a^2$. We know that the trinomial is $x^2 + 18x$, so we can set up the equation $x^2 + 18x = x^2 + 2ax + a^2$. Solving for $a$, we get $a = 9$. Therefore, the missing number is 81.

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

A: To determine if a trinomial is a perfect square, look at the formula $(x + a)^2 = x^2 + 2ax + a^2$ or $(x - a)^2 = x^2 - 2ax + a^2$. If the trinomial matches one of these formulas, then it is a perfect square.

Q: What are some common mistakes to avoid when finding the missing number?

A: Some common mistakes to avoid when finding the missing number include:

• Not using the correct formula
• Not identifying the value of $a$ correctly
• Not solving for $a$ correctly
• Not checking if the trinomial is a perfect square

Q: How do I practice finding the missing number?

A: To practice finding the missing number, try the following:

• Use online resources or worksheets to practice finding the missing number
• Work with a partner or tutor to practice finding the missing number
• Use real-world examples to practice finding the missing number
• Take practice quizzes or tests to assess your understanding of finding the missing number

Conclusion

In conclusion, finding the missing number to make a trinomial a perfect square trinomial requires the use of the formula $(x - a)^2 = x^2 - 2ax + a^2$ or $(x + a)^2 = x^2 + 2ax + a^2$. By identifying the value of $a$ and solving for $a$, we can find the missing number. Remember to practice finding the missing number to become proficient in this skill.