Question

Use suitable identities to find the following products: 

(i) (x + 4) (x + 10) 

(ii) (x + 8) (x – 10) 

(iii) (3x + 4) (3x – 5) 

(iv) \((y^ 2 + \frac{3 }{ 2}) (y^ 2 – \frac{3 }{ 2}) \)

(v) (3 – 2x) (3 + 2x)

Answer 1

Evaluating Products Using Algebraic Identities

1. Formula Used: 

We will use the following identities:

• For a binomial product: \( (a + b)(a + c) = a^2 + (b + c)a + bc \)
• For the difference of squares: \( (a + b)(a - b) = a^2 - b^2 \)

 

(i) \( (x + 4)(x + 10) \)

Step 1: Apply the formula for the product of two binomials: \[ (x + 4)(x + 10) = x^2 + (4 + 10)x + 4 \times 10 \] \[ = x^2 + 14x + 40 \]

(ii) \( (x + 8)(x - 10) \)

Step 1: Apply the formula for the product of two binomials: \[ (x + 8)(x - 10) = x^2 + (8 - 10)x + 8 \times (-10) \] \[ = x^2 - 2x - 80 \]

(iii) \( (3x + 4)(3x - 5) \)

Step 1: Apply the formula for the product of two binomials: \[ (3x + 4)(3x - 5) = (3x)^2 + (4 - 5)3x + 4 \times (-5) \] \[ = 9x^2 - 3x - 20 \]

(iv) \( \left(y^2 + \frac{3}{2}\right)\left(y^2 - \frac{3}{2}\right) \)

Step 1: Apply the difference of squares identity: \[ \left(y^2 + \frac{3}{2}\right)\left(y^2 - \frac{3}{2}\right) = \left(y^2\right)^2 - \left(\frac{3}{2}\right)^2 \] \[ = y^4 - \frac{9}{4} \]

(v) \( (3 - 2x)(3 + 2x) \)

Step 1: Apply the difference of squares identity: \[ (3 - 2x)(3 + 2x) = 3^2 - (2x)^2 \] \[ = 9 - 4x^2 \]

Answer Summary:

• (i) \( (x + 4)(x + 10) = x^2 + 14x + 40 \)
• (ii) \( (x + 8)(x - 10) = x^2 - 2x - 80 \)
• (iii) \( (3x + 4)(3x - 5) = 9x^2 - 3x - 20 \)
• (iv) \( \left(y^2 + \frac{3}{2}\right)\left(y^2 - \frac{3}{2}\right) = y^4 - \frac{9}{4} \)
• (v) \( (3 - 2x)(3 + 2x) = 9 - 4x^2 \)