Answered

Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Solving Quadratic Equations Using Algebra

Which is true about all quadratic equations that contain a difference of squares?

A. The value [tex]\(|b|=2 \sqrt{a} \sqrt{c}\)[/tex].
B. Only the value of [tex]\(c\)[/tex] is a perfect square.
C. Only the value of [tex]\(a\)[/tex] is a perfect square.
D. The value [tex]\(b=0\)[/tex].

Sagot :

GinnyAnswer
To determine the characteristics of quadratic equations that contain a difference of squares, let's begin by understanding the general form of such equations.

A quadratic equation typically has the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]

However, a quadratic equation that contains a difference of squares can be written in a different specific form, which is:
[tex]\[ a(x^2) - c = 0 \][/tex]

This form can be derived by recognizing the structure of the difference of squares:
[tex]\[ (x^2 - a^2) = (x + a)(x - a) \][/tex]

But in a more general term considering the equation [tex]\( a(x^2) - c = 0 \)[/tex], we note the following:
1. There is no [tex]\(bx\)[/tex] term, which implies that [tex]\( b = 0 \)[/tex].

Based on this structure:
[tex]\[ a(x^2) - c = 0 \][/tex]

The middle term ([tex]\(bx\)[/tex]) is missing; hence, [tex]\( b \)[/tex] has to be zero ([tex]\( b = 0 \)[/tex]).

This tells us something special about the value of [tex]\( b \)[/tex] in such quadratic equations:
- The value [tex]\( b \)[/tex] must be [tex]\( 0 \)[/tex].

Now, reviewing the given options:

1. The value [tex]\(|b| = 2 \sqrt{a} \sqrt{c}\)[/tex]: This statement is not true given our structure. The value of [tex]\( b \)[/tex] in the standard form is derived to be 0, irrespective of the values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex].
2. Only the value of [tex]\( c \)[/tex] is a perfect square: This is not necessarily true. While [tex]\( c \)[/tex] could be a perfect square, it is not a necessary condition for all quadratic equations of this type.
3. Only the value of [tex]\( a \)[/tex] is a perfect square: Similarly, [tex]\( a \)[/tex] being a perfect square is not a requirement for the general form of the equation [tex]\( a(x^2) - c = 0 \)[/tex].
4. The value [tex]\( b = 0 \)[/tex]: This aligns perfectly with our derived structure for the difference of squares equation. As shown, [tex]\( b \)[/tex] must be zero for the equation to lack the [tex]\( bx \)[/tex] term.

Therefore, the correct answer to the question is:
The value [tex]\( b = 0 \)[/tex].
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.