Question

Princess Poly says that the product of (74x+94i) and (74x−94i) is (4916x2+8116). Which statement is true regarding her product? Her product is not correct. The expression (74x+94i)(74x−94i) can be written as the difference of squares (74x)2−(94i)2, which is equal to 4916x2−8116. Her product is not correct. The expression (74x+94i)(74x−94i) can be written as the difference of squares (74x)2−(32i)2, which is equal to 4916x2−94. Her product is not correct. The expression (74x+94i)(74x−94i) can written as the difference of squares (74x)2−(32i)2, which is equal to 4916x2−94. Her product is correct. The expression (74x+94i)(74x−94i) can be written as the difference of squares (74x)2−(94i)2, which is equal to 4916x2+8116.

Answer 1

As you must know that (a + bi) and (a-bi) are complex numbers,each being conjugate of each other.

The meaning of conjugate is change the sign of the number preceding imaginary part.

In a+bi ,bi being the imaginary part .  It has positive sign before it .so just change the sign to -(negative) .So a-bi is the conjugate of a+bi. Similarly a+bi is the conjugate of a-bi.

Now com to the product of two complex number which are conjugate of each other.

(a+bi)(a-bi)=$a^{2} -(bi)^{2} \text [by  using  the  identity (a+b)(a-b)=a^{2}-b^{2}]$

                 =$a^2+b^2 [ because i^2=-1]$

So product of (74 x +94 i) and (74 x-94 i) which are conjugate of each other , their product will be

             $(74x)^2+(94)^2$

=$5476 x^2 + 8836$

The expression (74x+94i)(74x−94i) can be written as the difference of squares (74x)2−(94i)2, which is equal to 4916x2+8116.

This statement is true for Princess Poly Regarding her product of (74x+94i) and (74x−94i).

Answer 2

Solution:

(74x + 94i) x (74x - 94i)

can be written as (74x)² - (94i)²

which is equal to 5476x² + 8836

Given:

(74x + 94i) x (74x - 94i)

Formula Used:

Using the fact that $i^2 = - 1$

$(a + bi) . (a - bi) = (a)^2 - (bi)^2\\\\(a + bi) . (a - bi) = (a)^2 - (b)^2(i)^2                             \\\\(a + bi) . (a - bi) = (a)^2 -(-1) (b)^2\\\\(a+bi) . (a - bi) = (a)^2 + (b)^2$

Explanation:

Given the expression

$(74x + 94i) . (74x - 94i)\\= ( 74 x)^{2}  + (74x)(-94i) + (74x)(94i) -  (94i)^2\\=5476x^2 -8836.(-1)\\=5476x^2 + 8836$

Note:

The calculation suggests that 5476x² + 8836 is the final answer. Could 4916x²-8116 be a typo? Please update/confirm and I would be happy to provide any follow-ups required!