how to solve surd in maths?

Aliyuus2

6 months ago

To solve surds, you can either simplify them by finding the largest perfect square factor or solve surd equations by isolating and squaring the radical to eliminate the square root. For simplification, rewrite the number under the root as a product of a perfect square and another number, then take the square root of the perfect square out of the radical. For equations, isolate one surd on one side of the equation, square both sides, and repeat the process as needed. This video explains how to simplify surds step-by-step:00:52Breakthrough Maths YouTube • 8 Sept 2025Simplifying surds Find a perfect square factor: Find the largest perfect square (like 4, 9, 16, etc.) that divides the number inside the square root (the radicand).Example: For \(\sqrt{40}\), the largest perfect square factor is \(4\), since \(40=4\times 10\).Rewrite the surd: Split the radicand into the product of the perfect square and the other factor.Example: \(\sqrt{40}=\sqrt{4\times 10}\).Separate the square root: Use the rule \(\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\) to split the surd into two separate surds.Example: \(\sqrt{4\times 10}=\sqrt{4}\times \sqrt{10}\).Simplify the perfect square: Calculate the square root of the perfect square.Example: \(\sqrt{4}\times \sqrt{10}=2\sqrt{10}\).Combine and write the final answer: Write the simplified integer part before the remaining surd.Example: The simplified form of \(\sqrt{40}\) is \(2\sqrt{10}\). Solving surd equations Isolate the surd: Get the square root expression by itself on one side of the equation.Example: For the equation \(f=2+\sqrt{19-2f}\), subtract 2 from both sides to get \(f-2=\sqrt{19-2f}\).Square both sides: Square both sides of the equation to eliminate the square root.Example: \((f-2)^{2}=(\sqrt{19-2f})^{2}\), which simplifies to \(f^{2}-4f+4=19-2f\).Solve the resulting equation: Rearrange the terms to solve the new equation, which may be a quadratic equation.Example: \(f^{2}-2f-15=0\).Factor or use the quadratic formula: Solve for \(f\).Example: Factoring gives \((f-5)(f+3)=0\), so \(f=5\) or \(f=-3\).Check your solutions: Substitute the solutions back into the original equation to make sure they are valid. A solution is only valid if it works in the original equation.Example: Checking \(f=5\) gives \(5=2+\sqrt{19-10}\), which simplifies to \(5=2+\sqrt{9}=2+3=5\). This is true. Checking \(f=-3\) gives \(-3=2+\sqrt{19-2(-3)}=2+\sqrt{25}=2+5=7\). This is false, so \(f=-3\) is an extraneous solution.