In which visual situation is it most useful to read (x^2+6x+9) as (x^2+3x+3x+9)?
Correct answer: A
In a complete square model, (6x) is split into two equal (3x) rectangles. Exam tip: see the middle term as two equal rectangles.
View question detailsMathematics
Visual models of identities
सर्वसमिकाओं के दृश्य मॉडल
In Class 9 Mathematics, under the chapter Exploring Algebraic Identities, students use geometric diagrams and area-based representations to understand algebraic identities visually. They relate squares and rectangles formed from algebraic expressions to expansions such as binomial products, helping them see why both sides of an identity are equal rather than merely memorising formulas.
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Hard · Level 67 · complete-square,split-middle-term,visual-model
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- When it is changed into a complete square of side (x+3)
- When (9) is removed to make a line
- When (6x) is treated as one long strip
- When (x^2) is divided into two triangles
Hard · Level 67 · square-model,two-variable,visual-expansion
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- (25x^2+10xy+4y^2)
- (25x^2+20xy+4y^2)
- (5x^2+20xy+2y^2)
- (25x^2+4xy+20y^2)
Hard · Level 67 · missing-tiles,perfect-square,visual-identity
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- Two (ab) rectangles
- One (a+b) line
- One (a-b) rectangle
- Two (b^2) squares
Hard · Level 67 · signed-tiles,rectangle-model,factorization
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- ((x-4)(x+3))
- ((x+4)(x+3))
- ((x-4)(x-3))
- ((x+12)(x-1))
Hard · Level 67 · minus-binomial,positive-square,visual-model
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- (4a^2)
- (-4ab)
- (b^2)
- (-b^2)
Hard · Level 67 · algebraic identities, visual models, difference of squares, area model, class 9 mathematics
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- एक आयत, जिसकी भुजाएँ दो वर्गों की भुजाओं के योग और अंतर के बराबर होती हैं
- एक वर्ग, जिसकी भुजा दोनों वर्गों की भुजाओं के योग के बराबर होती है
- एक त्रिभुज, जिसका आधार बड़े वर्ग की भुजा के बराबर होता है
- एक समांतर चतुर्भुज, जिसकी सभी भुजाएँ समान होती हैं
Hard · Level 67 · three-term-square,pair-rectangles,visual-model
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- Once
- Twice
- Thrice
- Four times
Hard · Level 67 · three-term-identity,ab-term,area-model
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- (ab)
- (a^2b^2)
- (2ab)
- (3ab)
Hard · Level 67 · rectangle-expansion,cross-terms,visual-multiplication
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- (8x)
- (11x)
- (6x)
- (12x)
Hard · Level 67 · middle-rectangles,binomial-square,area-model
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- (10x)
- (20x)
- (25x)
- (40x)
Hard · Level 67 · subtracted-strips,minus-square,visual-identity
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- From two subtracted (6x) rectangles
- From the small (36) square
- From the large (x^2) square
- Only from the side (x-6)
Hard · Level 67 · common-error,binomial-square,visual-correction
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- Seeing two (ab) rectangles in a square of side (a+b)
- Drawing a line of side (a-b)
- Making a circle from a square
- Treating the rectangle as one part only
Hard · Level 67 · algebraic identities,visual models,square of binomial,area model,class 9 mathematics
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- एक a² वर्ग, एक b² वर्ग और ab क्षेत्रफल वाले दो आयत
- एक a² वर्ग, एक b² वर्ग और ab क्षेत्रफल वाला एक आयत
- a² क्षेत्रफल वाले दो वर्ग और b² क्षेत्रफल वाले दो वर्ग
- a² क्षेत्रफल वाला एक वर्ग और ab क्षेत्रफल वाले तीन आयत
Hard · Level 67 · completing-square,area-model,visual-method
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- Square of area (18)
- Square of area (36)
- Square of area (81)
- Square of area (324)
Hard · Level 67 · conjugate-rectangles,difference-identity,visual-model
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- (9x^2+4)
- (9x^2-4)
- (6x^2-4)
- (9x^2-12x+4)
Hard · Level 67 · signed-rectangle,constant-term,visual-expansion
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- (10)
- (-10)
- (3)
- (-7)
Hard · Level 67 · perfect-square-trinomial,minus-square,visual-recognition
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- Square with side (2a-3b)
- Square with side (4a-9b)
- Rectangle with dimensions (2a+3b) and (2a-3b)
- Square with side (a-3b)
Hard · Level 67 · error-analysis,area-model,binomial-square
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- (a^2+2ab+b^2)
- (a^2+ab+b^2)
- (a^2-b^2)
- (a^2+ab)
Hard · Level 67 · outer-side,perfect-square,visual-model
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- (7x+2)
- (49x+2)
- (7x+4)
- (14x+2)
Hard · Level 67 · rectangle-parts,expansion,visual-algebra
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- (4x^2+11x+6)
- (4x^2+8x+6)
- (4x^2+3x+2)
- (5x^2+11x+6)
What will be the total area in the visual model of a square with side (5x+2y)?
Correct answer: B
The middle term is (2\times5x\times2y=20xy). Exam tip: double the product of the two side parts.
View question detailsIf the model of (a^2+2ab+b^2) makes a complete square, what is missing if only (a^2+b^2) is kept?
Correct answer: A
Two (ab) rectangles are needed to complete the square. Exam tip: square terms alone do not form the whole square.
View question detailsA rectangle model has parts (x^2), (-4x), (3x), and (-12). Which multiplication does it show?
Correct answer: A
The number parts are (-4) and (3), whose product is (-12) and whose (x)-terms sum to (-x). Exam tip: read rectangle parts with signs.
View question detailsIn the visual model of ((2a-b)^2), which part is finally added as positive?
Correct answer: C
In the subtraction model, the small square from (b) is added positively as (b^2). Exam tip: the square of any term is positive.
View question detailsIn the visual model of the difference of two squares, into what shape can the remaining L-shaped region be rearranged after removing the smaller square from the larger square?
Correct answer: A
Let the side lengths of the larger and smaller squares be a and b. The remaining L-shaped region can be rearranged into a rectangle with sides (a+b) and (a-b), so its area is a²-b². Exam tip: identify both rectangle sides first.
View question detailsIn a square of side (x+y+z), how many times will rectangles of area (xy) appear?
Correct answer: B
The pair (x) and (y) forms two equal (xy) rectangles. Exam tip: count two rectangles for each different pair.
View question detailsIf a three-part square has square parts (a^2), (b^2), (c^2) and paired rectangles, what is the total contribution of (ab) in ((a+b+c)^2)?
Correct answer: C
There are two equal rectangles made from (a) and (b), so the total is (2ab). Exam tip: double the paired rectangles.
View question detailsIn the rectangle model of ((2x+1)(3x+4)), what is the total area of the (x)-terms?
Correct answer: B
Here (2x\times4=8x) and (1\times3x=3x), so the total is (11x). Exam tip: add the cross rectangles.
View question detailsA square model has square corners (4x^2) and (25). If the model is ((2x+5)^2), what is the total area of the two rectangles?
Correct answer: B
The total area of the two rectangles is (2\times2x\times5=20x). Exam tip: combine the two rectangles to form the middle term.
View question detailsIf the visual model of ((x-6)^2) gives (x^2-12x+36), where does (-12x) come from?
Correct answer: A
The total area of the two removed strips is (6x+6x=12x). Exam tip: connect the negative middle term with removed rectangles.
View question detailsWhich visual model helps explain that ((a+b)^2) is not only (a^2+b^2)?
Correct answer: A
A square of side (a+b) has two (ab) parts along with (a^2) and (b^2). Exam tip: never omit the middle (2ab).
View question detailsIn the visual model of the square of a+b, into which parts is the large square divided?
Correct answer: A
The large square has side a+b. Splitting each side into a and b gives one a² square, one b² square, and two ab rectangles, so its area is a²+2ab+b². In exams, remember to count both middle ab regions.
View question detailsWhich square should be added to the area model of (x^2+18x) to make a complete square?
Correct answer: C
Split (18x) into two (9x) rectangles, so (9^2=81) is added. Exam tip: take half the coefficient of (x) and square it.
View question detailsIn a visual model, the area of ((3x+2)(3x-2)) will be equal to what?
Correct answer: B
This is conjugate multiplication, so middle terms cancel and ((3x)^2-2^2=9x^2-4) remains. Exam tip: identify ((A+B)(A-B)).
View question detailsIf the outer dimensions of a rectangle model are (x+5) and (x-2), what will be the constant corner among the four parts?
Correct answer: B
In the rectangle model, the outer dimensions are \(x+5\) and \(x-2\). When these are split into variable and constant parts, the corner made from the two constant pieces has side lengths \(5\) and \(-2\). Its signed area is therefore \(5\times(-2)=-10\). The negative sign is essential because one dimension contributes a negative constant.
Expanding the whole product confirms the same result: \((x+5)(x-2)=x^2-2x+5x-10=x^2+3x-10\). The final constant term comes from the constant corner, so option B, \(-10\), is correct. Option A ignores the negative sign, while 3 and
\(-7\) do not result from multiplying the constant parts. In an area model, always multiply corresponding pieces with their signs.
In which model can (4a^2-12ab+9b^2) be seen as a complete square?
Correct answer: A
Here (4a^2=(2a)^2), (9b^2=(3b)^2), and the middle term is (-2\times2a\times3b). Exam tip: match all three terms with the perfect square pattern.
View question detailsIn a square of side (a+b), if the two (ab) rectangles are mistakenly counted only once, what area will be written?
Correct answer: B
If one (ab) is missed, the middle term becomes (ab) instead of (2ab). Exam tip: count both equal rectangles.
View question detailsIf the visual model of (49x^2+28x+4) is a complete square, what will be the outer side?
Correct answer: A
Here (49x^2=(7x)^2), (4=2^2), and (28x=2\times7x\times2). Exam tip: confirm using the middle term.
View question detailsWhat is obtained by adding all parts in the rectangle model of ((4x+3)(x+2))?
Correct answer: A
The parts are (4x^2), (8x), (3x), and (6), giving (4x^2+11x+6). Exam tip: add like (x)-terms.
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