Complete The Expression So It Forms A Perfect-square Trinomial.$\[ X^2 - 5x + \square \\]

What are Perfect-Square Trinomials?

A perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. It is a trinomial of the form ${ax^2 + bx + c}$, where ${a = 1}$ and ${c = b^2/4}$. In other words, a perfect-square trinomial is a quadratic expression that can be written in the form ${(x + d)^2}$, where ${d}$ is a constant.

Why are Perfect-Square Trinomials Important?

Perfect-square trinomials are important in algebra because they can be factored into the square of a binomial, which makes it easier to solve quadratic equations. They are also used in various mathematical applications, such as solving systems of equations and finding the maximum or minimum value of a quadratic function.

How to Complete the Expression to Form a Perfect-Square Trinomial

To complete the expression ${x^2 - 5x + \square}$ to form a perfect-square trinomial, we need to find the value of the missing term. We can do this by using the formula for the square of a binomial:

${(x + d)^2 = x^2 + 2dx + d^2}$

In this case, we have:

${x^2 - 5x + \square = (x + d)^2}$

We can equate the two expressions and solve for ${d}$:

${x^2 - 5x + \square = x^2 + 2dx + d^2}$

Subtracting ${x^2}$ from both sides gives:

${-5x + \square = 2dx + d^2}$

Equating the coefficients of ${x}$ gives:

${-5 = 2d}$

Solving for ${d}$ gives:

${d = -\frac{5}{2}}$

Now that we have found the value of ${d}$, we can substitute it into the expression for the square of a binomial:

${(x + d)^2 = (x - \frac{5}{2})^2}$

Expanding the square gives:

${(x - \frac{5}{2})^2 = x^2 - 5x + \frac{25}{4}}$

Therefore, the missing term in the expression ${x^2 - 5x + \square}$ is ${\frac{25}{4}}$.

Example 1: Completing the Expression to Form a Perfect-Square Trinomial

Find the missing term in the expression ${x^2 + 7x + \square}$.

Using the formula for the square of a binomial, we have:

${(x + d)^2 = x^2 + 2dx + d^2}$

In this case, we have:

${x^2 + 7x + \square = (x + d)^2}$

We can equate the two expressions and solve for ${d}$:

${x^2 + 7x + \square = x^2 + 2dx + d^2}$

Subtracting ${x^2}$ from both sides gives:

${7x + \square = 2dx + d^2}$

Equating the coefficients of ${x}$ gives:

${7 = 2d}$

Solving for ${d}$ gives:

${d = \frac{7}{2}}$

Now that we have found the value of ${d}$, we can substitute it into the expression for the square of a binomial:

${(x + d)^2 = (x + \frac{7}{2})^2}$

Expanding the square gives:

${(x + \frac{7}{2})^2 = x^2 + 7x + \frac{49}{4}}$

Therefore, the missing term in the expression ${x^2 + 7x + \square}$ is ${\frac{49}{4}}$.

Example 2: Completing the Expression to Form a Perfect-Square Trinomial

Find the missing term in the expression ${x^2 - 9x + \square}$.

Using the formula for the square of a binomial, we have:

${(x + d)^2 = x^2 + 2dx + d^2}$

In this case, we have:

${x^2 - 9x + \square = (x + d)^2}$

We can equate the two expressions and solve for ${d}$:

${x^2 - 9x + \square = x^2 + 2dx + d^2}$

Subtracting ${x^2}$ from both sides gives:

${-9x + \square = 2dx + d^2}$

Equating the coefficients of ${x}$ gives:

${-9 = 2d}$

Solving for ${d}$ gives:

${d = -\frac{9}{2}}$

Now that we have found the value of ${d}$, we can substitute it into the expression for the square of a binomial:

${(x + d)^2 = (x - \frac{9}{2})^2}$

Expanding the square gives:

${(x - \frac{9}{2})^2 = x^2 - 9x + \frac{81}{4}}$

Therefore, the missing term in the expression ${x^2 - 9x + \square}$ is ${\frac{81}{4}}$.

Conclusion

Q: What is a perfect-square trinomial?

A: A perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. It is a trinomial of the form ${ax^2 + bx + c}$, where ${a = 1}$ and ${c = b^2/4}$.

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

A: To determine if a trinomial is a perfect-square trinomial, you can use the formula for the square of a binomial:

${(x + d)^2 = x^2 + 2dx + d^2}$

If the trinomial can be written in this form, then it is a perfect-square trinomial.

Q: How do I complete the expression to form a perfect-square trinomial?

A: To complete the expression to form a perfect-square trinomial, you need to find the value of the missing term. You can do this by using the formula for the square of a binomial and equating the coefficients of the terms.

Q: What is the formula for the square of a binomial?

A: The formula for the square of a binomial is:

${(x + d)^2 = x^2 + 2dx + d^2}$

Q: How do I find the value of d in the formula for the square of a binomial?

A: To find the value of d, you can equate the coefficients of the terms in the trinomial and the formula for the square of a binomial. For example, if the trinomial is ${x^2 + 7x + \square}$, you can equate the coefficients of x to get:

${7 = 2d}$

Solving for d gives:

${d = \frac{7}{2}}$

Q: What is the missing term in the expression ${x^2 - 5x + \square}$?

A: The missing term in the expression ${x^2 - 5x + \square}$ is ${\frac{25}{4}}$.

Q: What is the missing term in the expression ${x^2 + 7x + \square}$?

A: The missing term in the expression ${x^2 + 7x + \square}$ is ${\frac{49}{4}}$.

Q: What is the missing term in the expression ${x^2 - 9x + \square}$?

A: The missing term in the expression ${x^2 - 9x + \square}$ is ${\frac{81}{4}}$.

Q: How do I use perfect-square trinomials to solve quadratic equations?

A: Perfect-square trinomials can be used to solve quadratic equations by factoring the trinomial into the square of a binomial. For example, if the quadratic equation is ${x^2 + 7x + 12 = 0}$, you can factor the trinomial as:

${(x + 3)(x + 4) = 0}$

Solving for x gives:

${x + 3 = 0 \quad \text{or} \quad x + 4 = 0}$

${x = -3 \quad \text{or} \quad x = -4}$

Q: How do I use perfect-square trinomials to find the maximum or minimum value of a quadratic function?

A: Perfect-square trinomials can be used to find the maximum or minimum value of a quadratic function by factoring the trinomial into the square of a binomial. For example, if the quadratic function is ${f(x) = x^2 + 7x + 12}$, you can factor the trinomial as:

${(x + 3)(x + 4) = 0}$

The maximum or minimum value of the function occurs when x is equal to the value that makes the binomial equal to zero. In this case, the maximum or minimum value occurs when x is equal to -3 or -4.

Conclusion

In this article, we have provided a Q&A guide to perfect-square trinomials. We have discussed the definition of a perfect-square trinomial, how to determine if a trinomial is a perfect-square trinomial, and how to complete the expression to form a perfect-square trinomial. We have also provided examples to illustrate the concept and discussed how to use perfect-square trinomials to solve quadratic equations and find the maximum or minimum value of a quadratic function.