Find the Value Using Identity Find the Value \[ x^2 + \frac{1}{x^2} = 66 \] Find: \[ x – \frac{1}{x} \] Solution: Using identity: \[ \left(x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 66-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 64 \] \[ x-\frac{1}{x} = \pm 8 \] Next Question / Full Exercise

Simplify Products Using Identity Simplify the Following Products \[ \left(m + \frac{n}{7}\right)^3 \left(m – \frac{n}{7}\right) \] Solution: \[ \left(m + \frac{n}{7}\right)^3 \left(m – \frac{n}{7}\right) \] \[ = \left(m + \frac{n}{7}\right)^2 \left(m + \frac{n}{7}\right) \left(m – \frac{n}{7}\right) \] Using identity: \[ (a+b)(a-b)=a^2-b^2 \] \[ = \left(m + \frac{n}{7}\right)^2 \left(m^2 – \frac{n^2}{49}\right) \] Again using identity: \[

Simplify Products Using Identity Simplify the Following Products \[ \left(\frac{1}{2}a – 3b\right) \left(3b + \frac{1}{2}a\right) \left(\frac{1}{4}a^2 + 9b^2\right) \] Solution: Using identity: \[ (a-b)(a+b)=a^2-b^2 \] \[ \left(\frac{1}{2}a – 3b\right) \left(3b + \frac{1}{2}a\right) = \left(\frac{1}{2}a\right)^2 – (3b)^2 \] \[ = \frac{1}{4}a^2 – 9b^2 \] Now the expression becomes: \[ \left(\frac{1}{4}a^2 – 9b^2\right) \left(\frac{1}{4}a^2 + 9b^2\right) \]

Find the Value Using Identity Find the Value \[ 3x – 7y = 10 \] \[ xy = -1 \] Find: \[ 9x^2 + 49y^2 \] Solution: Using identity: \[ (a-b)^2 = a^2 + b^2 – 2ab \] \[ (3x-7y)^2 = (3x)^2 + (7y)^2 – 2(3x)(7y) \] \[ = 9x^2 + 49y^2 – 42xy \]

Find the Value Using Identity Find the Value \[ 2x + 3y = 8 \] \[ xy = 2 \] Find: \[ 4x^2 + 9y^2 \] Solution: Using identity: \[ (a+b)^2 = a^2 + b^2 + 2ab \] \[ (2x+3y)^2 = (2x)^2 + (3y)^2 + 2(2x)(3y) \] \[ = 4x^2 + 9y^2 + 12xy \]

Find the Value Using Identity Find the Value \[ 9x^2 + 25y^2 = 181 \] \[ xy = -6 \] Find: \[ 3x + 5y \] Solution: Using identity: \[ (a+b)^2 = a^2 + b^2 + 2ab \] \[ (3x+5y)^2 = (3x)^2 + (5y)^2 + 2(3x)(5y) \] \[ = 9x^2 + 25y^2 + 30xy \]

Find the Values Using Identity Find the Values \[ x + \frac{1}{x} = \sqrt{5} \] Find: \[ x^2 + \frac{1}{x^2} \quad \text{and} \quad x^4 + \frac{1}{x^4} \] Solution: Using identity: \[ \left(x+\frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \] \[ (\sqrt{5})^2 = x^2 + \frac{1}{x^2} + 2 \] \[ 5 = x^2 + \frac{1}{x^2} +

Find the Value Using Identity Find the Value \[ x – \frac{1}{x} = -1 \] Find: \[ x^2 + \frac{1}{x^2} \] Solution: Using identity: \[ \left(x-\frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} – 2 \] \[ (-1)^2 = x^2 + \frac{1}{x^2} – 2 \] \[ 1 = x^2 + \frac{1}{x^2} – 2 \] \[ x^2 + \frac{1}{x^2}

Find the Value Using Identity Find the Value \[ x + \frac{1}{x} = 11 \] Find: \[ x^2 + \frac{1}{x^2} \] Solution: Using identity: \[ \left(x+\frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \] \[ (11)^2 = x^2 + \frac{1}{x^2} + 2 \] \[ 121 = x^2 + \frac{1}{x^2} + 2 \] \[ x^2 + \frac{1}{x^2}

Simplify Using Identity Simplify \[ \frac{7.83 \times 7.83 – 1.17 \times 1.17}{6.66} \] Solution: Using identity: \[ a^2 – b^2 = (a-b)(a+b) \] \[ \frac{(7.83)^2 – (1.17)^2}{6.66} \] \[ = \frac{(7.83 – 1.17)(7.83 + 1.17)}{6.66} \] \[ = \frac{(6.66)(9)}{6.66} \] \[ = 9 \] Next Question / Full Exercise