Tag Archives: Surds

Surd Equations Reducible to Quadratics

Surd Equations Reducible to Quadratics

Surd Equations Reducible to Quadratic for Math Algebra is done squaring both sides for removing surds and radical expressions. Make sure to check whether the solutions are correct by substituting them into the original surd equations. Question 1 Solve \( x = \sqrt{x+2} \). \( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}x^2 &= x+2 &\color{red} \text{square both sides} […]

Radical Indefinite Integrals

Radical Indefinite Integrals

Radical Indefinite Integrals should be performed after converting its radical or surd notations into index form.$$ \large \displaystyle \sqrt[n]{x^m} = x^{\frac{m}{n}}$$ Practice Questions Question 1 Find \( \displaystyle \int{\sqrt{x}}dx \). \( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}\int{\sqrt{x}}dx &= \int{x^{\frac{1}{2}}}dx &\color{red} \text{convert to index form} \\&= \dfrac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C \\&= \dfrac{x^{\frac{3}{2}}}{\frac{3}{2}} + C &\color{red} \text{ensure to convert back […]

Rationalising Denominators of Multiple Fractions

Rationalising Denominators of Multiple Fractions

Worked Example of Rationalising Denominators Simplify \( \displaystyle\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}} \). \( \begin{aligned} \displaystyle \require{color}&= \frac{1}{\sqrt{1} + \sqrt{2}} \times \frac{\sqrt{1}-\sqrt{2}}{\sqrt{1}-\sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}} \times \frac{\sqrt{99}-\sqrt{100}}{\sqrt{99}-\sqrt{100}} \\&= \frac{\sqrt{1}-\sqrt{2}}{1-2} + \frac{\sqrt{2}-\sqrt{3}}{2-3} + \cdots + \frac{\sqrt{99}-\sqrt{100}}{99-100} \\&= -(\sqrt{1}-\sqrt{2})-(\sqrt{2}-\sqrt{3})-\cdots-(\sqrt{99}-\sqrt{100}) \\&= […]