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For handling inequalities involving absolute values and surds, it is required to ensure the domains before solving inequalities. the final solutions must fit in the domains. Example 1 Solve for \( x \), \( |x| \gt \sqrt{x+2} \). \( \begin{align} x+2 &\ge 0 &\color{green}{\text{domain of } \sqrt{x+2} } \\ x &\ge -2 \color{green}{\cdots (1)} \\ […]
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 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 […]
Solving radical equations are required to isolate the radicals or surds to one side of the equations. Then square both sides. Here we have two important checkpoints. Checkpoint 1 for Solving radical equations Make sure the whole of both sides is to be squared but not squaring individual terms.This is what I say,$$ \large \begin{align} […]
For solving Quadratic Equations in Square Roots, it is required to square both sides entirely, but not individually. For instance,$$ \begin{align} (1 + 2)^2 &= 3^2 \\1^2 + 2^2 &\ne 3^2 \end{align} $$It is important to check the solutions to see if they work for the original equation if the original equation is squared. Worked […]
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}) \\&= […]
Nested square roots or nested radical problems are quite interesting to solve. The key skill for this question is to understand how the students can handle “…”. This enables us to set up a quadratic equation to evaluate its exact value using the quadratic formula,$$x= \frac{-b \ \pm \sqrt{b^2-4ac}}{2a}$$.Let’s look at the following examples for […]