Mathematics 2019 JAMB Past Questions - Question 8: Simplify 1/√3-2 – 1/√3+2

Correct Answer

D

Explanation

To simplify the expression 1/√3-2 – 1/√3+2, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b).Let's start by simplifying the denominators:To simplify the expression \( \frac{1}{\sqrt{3}-2} - \frac{1}{\sqrt{3}+2} \), we can use the difference of squares to rationalize the denominators.First, we rationalize the denominators by multiplying the numerators and denominators of each fraction by the conjugate of the denominator:For the first fraction:\[ \frac{1}{\sqrt{3}-2} \times \frac{\sqrt{3}+2}{\sqrt{3}+2} = \frac{\sqrt{3}+2}{(\sqrt{3}-2)(\sqrt{3}+2)} \]For the second fraction:\[ \frac{1}{\sqrt{3}+2} \times \frac{\sqrt{3}-2}{\sqrt{3}-2} = \frac{\sqrt{3}-2}{(\sqrt{3}+2)(\sqrt{3}-2)} \]Now, we can simplify the expression:\[ \frac{\sqrt{3}+2}{(\sqrt{3}-2)(\sqrt{3}+2)} - \frac{\sqrt{3}-2}{(\sqrt{3}+2)(\sqrt{3}-2)} \]The denominators cancel each other out, leaving us with:\[ \frac{\sqrt{3}+2 - (\sqrt{3}-2)}{3-4} = \frac{\sqrt{3}+2 - \sqrt{3}+2}{-1} = \frac{4}{-1} = -4 \]So, the simplified value of the expression is -4.√3 - 2 can be rationalized by multiplying the numerator and denominator by its conjugate, which is √3 + 2.Similarly, √3 + 2 can be rationalized by multiplying the numerator and denominator by its conjugate, which is √3 - 2.After rationalizing the denominators, the expression becomes:1/(√3 - 2) - 1/(√3 + 2)Now, let's rationalize the denominators:1/(√3 - 2) = (1/(√3 - 2)) * ((√3 + 2)/(√3 + 2)) = (√3 + 2)/(3 - 4) = -(√3 + 2)1/(√3 + 2) = (1/(√3 + 2)) * ((√3 - 2)/(√3 - 2)) = (√3 - 2)/(3 - 4) = -(√3 - 2)So, the expression simplifies to:-(√3 + 2) - (-√3 + 2) = -√3 - 2 + √3 - 2 = -4Therefore, the simplified value of the expression 1/√3-2 – 1/√3+2 is -4.