If $x^2 + Mx + M$ Is A Perfect-square Trinomial, Which Equation Must Be True?A. $x^2 + Mx + M = (x - 1)^2$B. \$x^2 + Mx + M = (x + 1)^2$[/tex\]C. $x^2 + Mx + M = (x + 2)^2$D. $x^2 + Mx + M = (x +

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If $x^2 + mx + m$ is a Perfect-Square Trinomial, Which Equation Must Be True?

Understanding Perfect-Square Trinomials

A perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form of $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant. In this case, we are given the quadratic expression $x^2 + mx + m$ and asked to determine which equation must be true if it is a perfect-square trinomial.

The General Form of a Perfect-Square Trinomial

The general form of a perfect-square trinomial is given by $(x + a)^2$ or $(x - a)^2$. When we expand these expressions, we get:

(x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2

(x−a)2=x2−2ax+a2(x - a)^2 = x^2 - 2ax + a^2

Comparing with the Given Expression

Now, let's compare the given expression $x^2 + mx + m$ with the general form of a perfect-square trinomial. We can see that the constant term in the given expression is $m$, which is the same as the constant term in the general form of a perfect-square trinomial.

Determining the Value of $m$

To determine the value of $m$, we need to find the value of $a$ in the general form of a perfect-square trinomial. We can do this by comparing the linear term in the given expression with the linear term in the general form of a perfect-square trinomial.

Case 1: $x^2 + mx + m = (x + a)^2$

In this case, the linear term in the given expression is $mx$, which is equal to $2ax$ in the general form of a perfect-square trinomial. Therefore, we can write:

mx=2axmx = 2ax

m=2am = 2a

Case 2: $x^2 + mx + m = (x - a)^2$

In this case, the linear term in the given expression is $mx$, which is equal to $-2ax$ in the general form of a perfect-square trinomial. Therefore, we can write:

mx=−2axmx = -2ax

m=−2am = -2a

Determining the Value of $a$

Now that we have the value of $m$ in terms of $a$, we can determine the value of $a$ by substituting the value of $m$ into one of the equations.

Case 1: $m = 2a$

Substituting $m = 2a$ into the equation $x^2 + mx + m = (x + a)^2$, we get:

x2+2ax+2a=(x+a)2x^2 + 2ax + 2a = (x + a)^2

Expanding the right-hand side of the equation, we get:

x2+2ax+2a=x2+2ax+a2x^2 + 2ax + 2a = x^2 + 2ax + a^2

Subtracting $x^2 + 2ax$ from both sides of the equation, we get:

2a=a22a = a^2

Subtracting $a$ from both sides of the equation, we get:

a=a2−aa = a^2 - a

Factoring the right-hand side of the equation, we get:

a=a(a−1)a = a(a - 1)

Dividing both sides of the equation by $(a - 1)$, we get:

a=aa = a

This equation is true for all values of $a$, but it does not provide any information about the value of $a$.

Case 2: $m = -2a$

Substituting $m = -2a$ into the equation $x^2 + mx + m = (x - a)^2$, we get:

x2−2ax−2a=(x−a)2x^2 - 2ax - 2a = (x - a)^2

Expanding the right-hand side of the equation, we get:

x2−2ax−2a=x2−2ax+a2x^2 - 2ax - 2a = x^2 - 2ax + a^2

Subtracting $x^2 - 2ax$ from both sides of the equation, we get:

−2a=a2-2a = a^2

Subtracting $a$ from both sides of the equation, we get:

−2a−a=a2-2a - a = a^2

Simplifying the left-hand side of the equation, we get:

−3a=a2-3a = a^2

Subtracting $a^2$ from both sides of the equation, we get:

−3a−a2=0-3a - a^2 = 0

Factoring the left-hand side of the equation, we get:

−a(3+a)=0-a(3 + a) = 0

This equation is true when $a = 0$ or $a = -3$.

Conclusion

In conclusion, if $x^2 + mx + m$ is a perfect-square trinomial, then the equation $x^2 + mx + m = (x + 2)^2$ must be true.

Answer

The correct answer is C. $x^2 + mx + m = (x + 2)^2$
Q&A: 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 $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant.

Q: How do I determine if a quadratic expression is a perfect-square trinomial?

A: To determine if a quadratic expression is a perfect-square trinomial, you need to check if it can be factored into the square of a binomial. You can do this by looking for a pattern in the expression, such as a perfect square in the constant term.

Q: What is the general form of a perfect-square trinomial?

A: The general form of a perfect-square trinomial is given by $(x + a)^2$ or $(x - a)^2$. When we expand these expressions, we get:

(x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2

(x−a)2=x2−2ax+a2(x - a)^2 = x^2 - 2ax + a^2

Q: How do I compare a quadratic expression with the general form of a perfect-square trinomial?

A: To compare a quadratic expression with the general form of a perfect-square trinomial, you need to look at the constant term and the linear term in the expression. If the constant term is a perfect square and the linear term is twice the product of the constant term and the variable, then the expression is a perfect-square trinomial.

Q: What is the value of $m$ in the expression $x^2 + mx + m$?

A: The value of $m$ in the expression $x^2 + mx + m$ is equal to $2a$ or $-2a$, where $a$ is a constant.

Q: How do I determine the value of $a$ in the expression $x^2 + mx + m$?

A: To determine the value of $a$ in the expression $x^2 + mx + m$, you need to substitute the value of $m$ into one of the equations and solve for $a$.

Q: What is the correct equation if $x^2 + mx + m$ is a perfect-square trinomial?

A: The correct equation if $x^2 + mx + m$ is a perfect-square trinomial is $x^2 + mx + m = (x + 2)^2$.

Q: What are some common mistakes to avoid when working with perfect-square trinomials?

A: Some common mistakes to avoid when working with perfect-square trinomials include:

  • Not recognizing the pattern of a perfect square in the constant term
  • Not checking if the linear term is twice the product of the constant term and the variable
  • Not substituting the value of $m$ into one of the equations to solve for $a$
  • Not factoring the expression correctly

Q: How do I practice working with perfect-square trinomials?

A: To practice working with perfect-square trinomials, you can try the following:

  • Start with simple expressions and gradually move on to more complex ones
  • Practice factoring expressions into the square of a binomial
  • Try to identify the pattern of a perfect square in the constant term
  • Use online resources or worksheets to practice working with perfect-square trinomials

Conclusion

In conclusion, perfect-square trinomials are an important concept in algebra that can be used to factor quadratic expressions into the square of a binomial. By understanding the general form of a perfect-square trinomial and how to compare it with a quadratic expression, you can determine if an expression is a perfect-square trinomial and factor it correctly. Remember to practice working with perfect-square trinomials to become more confident and proficient in your algebra skills.