Math Problem Statement
Which will result in a perfect square trinomial? (3 x minus 5)(3 x minus 5) (3 x minus 5)(5 minus 3 x) (3 x minus 5)(3 x + 5) (3 x minus 5)(negative 3 x minus 5)
Solution
To determineA which perfect product results in a perfect square trinomial square trin isomial, formed when we need a to binomial expand each is multiplied expression and by itself, see if the such as result matches (( the forma of + a b perfect)^ square2 trin =omial a^2, (a +^ 22ab \pm + b 2^ab +2\ b).^ Therefore2,\ we).
need### to determine1 which. option \ is(( a3 productx of - a bin5omial)( and3 itselfx.
-### 5 Step-by)-step analysis) This:
is1 the. square \ of(( a3 binxomial:
-\ [
5()(33xx - - 55))^):2
= ( This3 isx the)^ product of2 a - bin 2omial( and3 itselfx)(:
5 )[
- (53^x2 - =5 9)^x2^ =2 -9x 30^x2 + - 2530 x\ +] This25 is a perfect \ square] trin omial This.
is indeed---
a perfect2 square. trinomial ((.
32x. - (( 53)(x -5 - 53)(x5)\ -)
This3 isx a)\ product): of
two bin Simplomify usingials where distribut theive property terms:
are reversed :
\[
[
((33xx - - 55)()(55 - - 33xx)) = = (153xx)( -5 9) +x (^32x - )(-253 +x 15)x + (- = -59)(x5^)2 + (- +5 )(30-x3 -x 25)
15 \x]
- This 9 isx not^ a2 perfect - square trin25omial +.
- \x(( =3x -9 -x ^5)(2 +3 x +30 x5 - )\25):
] SimplifyThis using is not the difference a perfect of square squares: trin omial because \ the[ leading term (3 is negativex - and the middle5 term)( does3x not match + the form5) of = ( 2ab9x^2).
- ###25 3 . \ ((] 3 x This - is5 not)( a3 trinomial.
x + 45.)\ (()
3This isx a - difference of5)( squares:
-\3[
x(3 - x5 - )):5
)( Simpl3xify + using distribut5ive property):
= ( \3[
x)^ 2 ( -3 (x5 - )^52 =)( -93xx^ -2 -5 )25 = -
\9]
xThis^ is2 not - a perfect15 squarex trin +omial ;15 itx is + a difference25 of = squares -.
9---
x^###2 4 +. 25
((3 \x]
- 5 This)( is- not3 a trinx -omial.
5###)\ Correct) answerThis: is\ a(( product3 ofx - two binom5ials)( with3 oppositex leading - coefficients: 5)[ )
(Would3 youx like further - explanation or5)( have- other questions3?x 😊 -
5###) = Related ( Questions3: x)(1.- What3 arex the general) + conditions ( for3 ax trin)(omial- to5 be) a + perfect (- square? 52)(-. How can3 wex) identify + a (-5 perfect square)(- trinomial5 in) = standard - form? 9x3.^ What2 - is the15 differencex between + a perfect15 squarex trin +omial and25 a difference= of - squares9? x4^.2 Can + \ ((a25 + \ b] )(aThis - is b not)\ a perfect) square ever result in trin aomial perfect; square it simplifies trin to aomial? difference of5. squares How with a can factoring negative leading confirm term whether.
a---
trin###omial ** isConclusion a** perfectThe square first?
option**,Tip:** To ((3 quickly checkx - if a5 trin)(omial3 is ax perfect - square,5)\ verify if), the results in middle term a is perfect twice the square trin productomial: of the\ square[ roots9x of the^2 first - and last terms30x. + 25 ]
Would you like further clarification or a deeper explanation of perfect square trinomials? Here are 5 follow-up questions for practice:
- How can you identify a perfect square trinomial by looking at its expanded form?
- Expand ((2x + 7)^2) and verify if it's a perfect square trinomial.
- What is the relationship between the middle term and the square roots of the first and last terms in a perfect square trinomial?
- Can the product of two distinct binomials ever result in a perfect square trinomial? Why or why not?
- What is the general formula for expanding ((ax + b)^2)?
Tip: A perfect square trinomial always has the middle term as twice the product of the square roots of the first and last terms.
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Math Problem Analysis
Mathematical Concepts
Algebra
Factoring
Perfect Square Trinomials
Formulas
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
Theorems
Perfect Square Theorem
Suitable Grade Level
Grades 8-10
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