Simplify the following expression:$(2z - 5y)^2 + (5z + 2y)^2 ... | Mathematics MCQ | Examoogle

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Simplify the following expression: $(2z - 5y)^2 + (5z + 2y)^2 - 25z^2$

$19y^2 + 4z^2$

$19y^2 - 4z^2$

$29y^2 + 4z^2$

$29y^2 - 4z^2$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option C

$29y^2 + 4z^2$

Quick Summary:

Expand each binomial using standard algebraic identities, then combine like terms directly without complex substitution.

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📐MAMath SolutionDirect Formula

📋 Given

Expression: $(2z - 5y)^2 + (5z + 2y)^2 - 25z^2$

🔢 Formula Used

(a−b)2=a2−2ab+b2,(a+b)2=a2+2ab+b2(a - b)^2 = a^2 - 2ab + b^2, \quad (a + b)^2 = a^2 + 2ab + b^2(a−b)2=a2−2ab+b2,(a+b)2=a2+2ab+b2

⚡ Exam Hall Shortcut / Speed Trick

Expand each binomial using standard algebraic identities, then combine like terms directly without complex substitution.

⚠️ Common Student Trap / Pitfall

Students often miscalculate the middle terms $(2 \times 2z \times 5y)$ or drop negative signs during expansion.

🔢 Step-by-Step Solution

Step 1: Expand the first term

Expand $(2z - 5y)^2$ using the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$ .

(2z−5y)2=4z2−20yz+25y2(2z - 5y)^2 = 4z^2 - 20yz + 25y^2(2z−5y)2=4z2−20yz+25y2

Step 2: Expand the second term

Expand $(5z + 2y)^2$ using the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$ .

(5z+2y)2=25z2+20yz+4y2(5z + 2y)^2 = 25z^2 + 20yz + 4y^2(5z+2y)2=25z2+20yz+4y2

Step 3: Combine all terms

Substitute the expanded forms into the original expression along with $- 25z^2$ .

(4z2−20yz+25y2)+(25z2+20yz+4y2)−25z2(4z^2 - 20yz + 25y^2) + (25z^2 + 20yz + 4y^2) - 25z^2(4z2−20yz+25y2)+(25z2+20yz+4y2)−25z2

Step 4: Simplify and cancel like terms

Cancel $-20yz$ and $+20yz$ , and combine $4z^2 + 25z^2 - 25z^2 = 4z^2$ . Combine $25y^2 + 4y^2 = 29y^2$ .

29y2+4z229y^2 + 4z^229y2+4z2

✅

C is correct because expanding and simplifying the given expression yields $29y^2 + 4z^2$ .

Core Concepts Used

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Algebraic Identities Binomial Expansion Combining Like Terms

💡 EXAM TIP

Mastering algebraic expansions helps save time in polynomial simplification questions across quantitative aptitude.

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