What Values Are Needed To Make Each Expression A Perfect Square Trinomial?1. \[$x^2 + 2x +\$\] \[$\square\$\]2. \[$x^2 - 20x +\$\] \[$\square\$\]3. \[$x^2 + 5x +\$\] \[$\square\$\]

What Values are Needed to Make Each Expression a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form of ${(a+b)^2}$ or ${(a-b)^2}$, where ${a}$ and ${b}$ are constants. In this article, we will explore the values needed to make each expression a perfect square trinomial.

Understanding Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written in the form of ${(a+b)^2}$ or ${(a-b)^2}$. It has the following form:

${(a+b)^2 = a^2 + 2ab + b^2}$

or

${(a-b)^2 = a^2 - 2ab + b^2}$

where ${a}$ and ${b}$ are constants.

Making Each Expression a Perfect Square Trinomial

To make each expression a perfect square trinomial, we need to find the values of ${a}$ and ${b}$ that satisfy the given expression.

1. ${x^2 + 2x + \square}$

To make this expression a perfect square trinomial, we need to find the value of ${b}$ that satisfies the equation:

${2x = 2ab}$

where ${a = x}$ and ${b}$ is a constant.

Solving for ${b}$, we get:

${b = \frac{2x}{2a}}$

${b = \frac{x}{a}}$

Since ${a = x}$, we can substitute this value into the equation:

${b = \frac{x}{x}}$

${b = 1}$

Now that we have found the value of ${b}$, we can substitute it back into the original expression:

${x^2 + 2x + 1}$

This expression can be factored into the square of a binomial:

${(x+1)^2}$

Therefore, the value needed to make the expression ${x^2 + 2x + \square}$ a perfect square trinomial is ${b = 1}$.

2. ${x^2 - 20x + \square}$

To make this expression a perfect square trinomial, we need to find the value of ${b}$ that satisfies the equation:

${-20x = 2ab}$

where ${a = x}$ and ${b}$ is a constant.

Solving for ${b}$, we get:

${b = \frac{-20x}{2a}}$

${b = \frac{-10x}{a}}$

Since ${a = x}$, we can substitute this value into the equation:

${b = \frac{-10x}{x}}$

${b = -10}$

Now that we have found the value of ${b}$, we can substitute it back into the original expression:

${x^2 - 20x + 100}$

This expression can be factored into the square of a binomial:

${(x-10)^2}$

Therefore, the value needed to make the expression ${x^2 - 20x + \square}$ a perfect square trinomial is ${b = 100}$.

3. ${x^2 + 5x + \square}$

To make this expression a perfect square trinomial, we need to find the value of ${b}$ that satisfies the equation:

${5x = 2ab}$

where ${a = x}$ and ${b}$ is a constant.

Solving for ${b}$, we get:

${b = \frac{5x}{2a}}$

${b = \frac{5x}{2x}}$

${b = \frac{5}{2}}$

Now that we have found the value of ${b}$, we can substitute it back into the original expression:

${x^2 + 5x + \frac{25}{4}}$

This expression can be factored into the square of a binomial:

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

Therefore, the value needed to make the expression ${x^2 + 5x + \square}$ a perfect square trinomial is ${b = \frac{25}{4}}$.

Conclusion

In this article, we have explored the values needed to make each expression a perfect square trinomial. We have found that the values of ${b}$ needed to make each expression a perfect square trinomial are ${b = 1}$, ${b = 100}$, and ${b = \frac{25}{4}}$. These values can be used to factor each expression into the square of a binomial.

References

• [1] Algebra, 2nd ed. by Michael Artin
• [2] Calculus, 3rd ed. by Michael Spivak
• [3] Mathematics, 2nd ed. by Richard Courant

Further Reading

• [1] Perfect Square Trinomials, Math Open Reference
• [2] Factoring Perfect Square Trinomials, Mathway
• [3] Perfect Square Trinomials, Khan Academy
  Frequently Asked Questions: Perfect Square Trinomials

In this article, we will answer some of the most frequently asked questions about perfect square trinomials.

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 has the form of ${(a+b)^2}$ or ${(a-b)^2}$, where ${a}$ and ${b}$ are constants.

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

A: To determine if an expression is a perfect square trinomial, you can use the following steps:

1. Check if the expression is in the form of ${(a+b)^2}$ or ${(a-b)^2}$.
2. Check if the middle term is twice the product of the square roots of the first and last terms.
3. Check if the last term is the square of the first term.

If all of these conditions are met, then the expression is a perfect square trinomial.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you can use the following steps:

1. Identify the square root of the first term.
2. Identify the square root of the last term.
3. Write the expression in the form of ${(a+b)^2}$ or ${(a-b)^2}$, where ${a}$ is the square root of the first term and ${b}$ is the square root of the last term.

Q: What are some examples of perfect square trinomials?

A: Some examples of perfect square trinomials include:

• ${(x+1)^2 = x^2 + 2x + 1}$
• ${(x-3)^2 = x^2 - 6x + 9}$
• ${(x+2)^2 = x^2 + 4x + 4}$

Q: Can a perfect square trinomial have a negative sign?

A: Yes, a perfect square trinomial can have a negative sign. For example:

• ${(x-1)^2 = x^2 - 2x + 1}$
• ${(x+3)^2 = x^2 + 6x + 9}$

Q: Can a perfect square trinomial have a fraction as a coefficient?

A: Yes, a perfect square trinomial can have a fraction as a coefficient. For example:

• ${(x+\frac{1}{2})^2 = x^2 + x + \frac{1}{4}}$
• ${(x-\frac{1}{3})^2 = x^2 - \frac{2}{3}x + \frac{1}{9}}$

Q: Can a perfect square trinomial have a negative fraction as a coefficient?

A: Yes, a perfect square trinomial can have a negative fraction as a coefficient. For example:

• ${(x-\frac{1}{2})^2 = x^2 + x + \frac{1}{4}}$
• ${(x+\frac{1}{3})^2 = x^2 + \frac{2}{3}x + \frac{1}{9}}$

Q: Can a perfect square trinomial have a decimal as a coefficient?

A: Yes, a perfect square trinomial can have a decimal as a coefficient. For example:

• ${(x+0.5)^2 = x^2 + x + 0.25}$
• ${(x-0.3)^2 = x^2 - 0.6x + 0.09}$

Q: Can a perfect square trinomial have a negative decimal as a coefficient?

A: Yes, a perfect square trinomial can have a negative decimal as a coefficient. For example:

• ${(x-0.5)^2 = x^2 + x + 0.25}$
• ${(x+0.3)^2 = x^2 + 0.6x + 0.09}$

Conclusion

In this article, we have answered some of the most frequently asked questions about perfect square trinomials. We have covered topics such as what a perfect square trinomial is, how to determine if an expression is a perfect square trinomial, and how to factor a perfect square trinomial. We have also provided examples of perfect square trinomials with different coefficients, including fractions and decimals.