For which value of k is x² + 14x + k a perfect square?

49. A perfect square looks like (x + b)² = x² + 2bx + b². Matching 2b = 14 gives b = 7, so k = b² = 49 and x² + 14x + 49 = (x + 7)². Stopping at b = 7 gives 7; squaring 14 instead of 7 gives 196.

For which value of $k$ is $x^{2} + 14 x + k$ a perfect square?

Class 8Algebraic Expressions and IdentitiesAlgebra

A49
B7
C14
D196

Step-by-Step Solution

A perfect square looks like

(x + b)^{2} = x^{2} + 2 b x + b^{2}

. Matching

2 b = 14

gives

b = 7

, so

k = b^{2} = 49

and

x^{2} + 14 x + 49 = (x + 7)^{2}

. Stopping at

b = 7

gives 7; squaring 14 instead of 7 gives 196.

Answer: A) 49

Key idea: Why identities matter

Identities save time and reveal structure. Examples of clever uses:

- Compute

10 2^{2}

mentally:

(100 + 2)^{2} = 10 0^{2} + 2 (100) (2) + 2^{2} = 10000 + 400 + 4 = 10404

.
- Compute

99 \times 101 : (100 - 1) (100 + 1) = 10 0^{2} - 1^{2} = 10000 - 1 = 9999

.
- Compute

4 7^{2} - 4 3^{2} : (47 - 43) (47 + 43) = 4 \times 90 = 360

.

Look at any number-pair calculation; if you can rewrite it as one of the three identity forms, the answer falls out.

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For which value of k is x2+14x+k a perfect square?

Step-by-Step Solution

Key idea: Why identities matter

Master this chapter in 30 minutes a day.

More Class 8 Algebraic Expressions and Identities practice

For which value of $k$ is $x^{2} + 14 x + k$ a perfect square?