Educational

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

Simplify Using Identity Simplify \[ 0.76 \times 0.76 + 2 \times 0.76 \times 0.24 + 0.24 \times 0.24 \] Solution: Using identity: \[ a^2 + 2ab + b^2 = (a+b)^2 \] \[ 0.76 \times 0.76 + 2 \times 0.76 \times 0.24 + 0.24 \times 0.24 \] \[ = (0.76)^2 + 2(0.76)(0.24) + (0.24)^2 \] \[

Simplify Using Identity Simplify \[ 322 \times 322 – 2 \times 322 \times 22 + 22 \times 22 \] Solution: Using identity: \[ a^2 – 2ab + b^2 = (a-b)^2 \] \[ 322 \times 322 – 2 \times 322 \times 22 + 22 \times 22 \] \[ = (322)^2 – 2(322)(22) + (22)^2 \] \[

Simplify Using Identity Simplify \[ 175 \times 175 + 2 \times 175 \times 25 + 25 \times 25 \] Solution: Using identity: \[ a^2 + 2ab + b^2 = (a+b)^2 \] \[ 175 \times 175 + 2 \times 175 \times 25 + 25 \times 25 \] \[ = (175)^2 + 2(175)(25) + (25)^2 \] \[

Evaluate Using Identity Evaluate Using Identity \[ 117 \times 83 \] Solution: Using identity: \[ (a+b)(a-b)=a^2-b^2 \] \[ 117 \times 83 = (100+17)(100-17) \] \[ = (100)^2 – (17)^2 \] \[ = 10000 – 289 \] \[ = 9711 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate Using Identity \[ 991 \times 1009 \] Solution: Using identity: \[ (a-b)(a+b)=a^2-b^2 \] \[ 991 \times 1009 = (1000-9)(1000+9) \] \[ = (1000)^2 – (9)^2 \] \[ = 1000000 – 81 \] \[ = 999919 \] Next Question / Full Exercise