Exploring Algebraic Identities | EOT

Question 5

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

(i) $25a^2 - 30ab + 9b^2$

(ii) $36s^2 - 49t^2$

Solution

Understand the Question

The area of a rectangle is given by $\text{Area} = \text{Length} \times \text{Breadth}$ .
To find possible expressions for the length and breadth, factorise each quadratic expression into a product of two linear factors.
Useful algebraic identities:
- $(x - y)^2 = x^2 - 2xy + y^2$
- $x^2 - y^2 = (x + y)(x - y)$

(i) $25a^2 - 30ab + 9b^2$

Step 1 · Factorise the Quadratic Expression

Given area of the rectangle: $\text{Area} = 25a^2 - 30ab + 9b^2$

Rewrite the expression in the form $x^2 - 2xy + y^2$ : $25a^2 - 30ab + 9b^2 = (5a)^2 - 2(5a)(3b) + (3b)^2$

Using the algebraic identity $(x - y)^2 = (x - y)(x - y)$ : $\text{Area} = (5a - 3b)^2 = (5a - 3b)(5a - 3b)$

Since $\text{Area} = \text{Length} \times \text{Breadth}$ : $\text{Length} = 5a - 3b, \quad \text{Breadth} = 5a - 3b$

Answer

(i) $\text{Length} = 5a - 3b, \quad \text{Breadth} = 5a - 3b$

(ii) $36s^2 - 49t^2$

Step 1 · Factorise using Difference of Squares

Given area of the rectangle: $\text{Area} = 36s^2 - 49t^2$

Rewrite the expression in the form $x^2 - y^2$ : $36s^2 - 49t^2 = (6s)^2 - (7t)^2$

Using the algebraic identity $x^2 - y^2 = (x + y)(x - y)$ : $\text{Area} = (6s + 7t)(6s - 7t)$

Since $\text{Area} = \text{Length} \times \text{Breadth}$ : $\text{Length} = 6s + 7t, \quad \text{Breadth} = 6s - 7t \quad (\text{or vice versa})$

Answer

(ii) $\text{Length} = 6s + 7t, \quad \text{Breadth} = 6s - 7t \quad (\text{or vice versa})$

Common Mistakes

Identity Confusion: Conflating the perfect square identity $(x - y)^2 = x^2 - 2xy + y^2$ with the difference of squares $x^2 - y^2 = (x - y)(x + y)$ .
Interchangeable Factors: Forgetting that length and breadth are interchangeable factors since multiplication is commutative.

More questions in EOT

Use suitable identities to find the following products:

(i) $(-3x + 4)^2$

(ii) $(2s + 7)(2s - 7)$

(iii) $\left(p^2 + \dfrac{1}{2}\right)\left(p^2 - \dfrac{1}{2}\right)$

(iv) $(2n + 7)(2n - 7)$

(v) $(s - 2t)(s^2 + 2st + 4t^2)$

(vi) $\left(\dfrac{1}{2r} - 4r\right)^2$

(vii) $(-3m + 4k - l)^2$

(viii) $\left(x - \dfrac{1}{3}y\right)^3$

(ix) $\left(\dfrac{7}{2}k - \dfrac{2}{3}m\right)^3$

Find the values using suitable identities:

(i) $17 \times 21$

(ii) $104 \times 96$

(iii) $24 \times 16$

(iv) $147^3$

(v) $199^3$

(vi) $127^3$

(vii) $(-107)^3$

(viii) $(-299)^3$

Factor the following algebraic expressions:

(i) $4y^2 + 1 + \dfrac{1}{16y^2}$

(ii) $9m^2 - \dfrac{1}{25n^2}$

(iii) $27b^3 - \dfrac{1}{64b^3}$

(iv) $x^2 + \dfrac{5x}{6} + \dfrac{1}{6}$

(v) $27u^3 - \dfrac{1}{125} - \dfrac{27u^2}{5} + \dfrac{9u}{25}$

(vi) $64y^3 + \dfrac{1}{125}z^3$

(vii) $p^3 + 27q^3 + r^3 - 9pqr$

(viii) $9m^2 - 12m + 4$

(ix) $9x^3 - \dfrac{8}{3}y^3 + \dfrac{z^3}{3} + 6xyz$

(x) $4x^2 + 9y^2 + 36z^2 + 12xz + 36yz + 24xy$

(xi) $27u^3 - \dfrac{1}{216} - \dfrac{9u^2}{2} + \dfrac{u}{4}$

Simplify the following:

(i) $\dfrac{4x^2 + 4x + 1}{4x^2 - 1}$

(ii) $\dfrac{9(3a^3 - 24b^3)}{9a^2 - 36b^2}$

(iii) $\dfrac{s^3 + 125t^3}{s^2 - 2st - 35t^2}$

Note: Assume that the denominators are not equal to 0.

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

(i) $25a^2 - 30ab + 9b^2$

(ii) $36s^2 - 49t^2$

Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.

(i) $6a^2 - 24b^2$

(ii) $3ps^2 - 15ps + 12p$

The village playground is shaped as a square of side 40 metres. A path of width $s$ metres is created around the playground for people to walk. Find an expression for the area of the path in terms of $s$ .

If a number plus its reciprocal equals $\dfrac{10}{3}$ , find the number.

A rectangular pool has area $2x^2 + 7x + 3$ square hastas. If its width is $2x + 1$ hastas, find its length. Hasta was a unit used to measure length.

Q10

If both $x - 2$ and $x - \dfrac{1}{2}$ are factors of $px^2 + 5x + r$ , show that $p = r$ .

Q11

If $a + b + c = 5$ and $ab + bc + ca = 10$ , then prove that $a^3 + b^3 + c^3 - 3abc = -25$ .

Q12

By factoring the expression, check that $n^3 - n$ is always divisible by 6 for all natural numbers $n$ . Give reasons.

Q13

Find the value of

(i) $x^3 + y^3 - 12xy + 64$ , when $x + y = -4$

(ii) $x^3 - 8y^3 - 36xy - 216$ , when $x = 2y + 6$