Amazing tricks on solving this radical equation скачать в хорошем качестве

Amazing tricks on solving this radical equation

After watching this video, you would be able to solve this challenging radical equation which has complex solutions. Radicals A radical is a mathematical expression that involves a root, such as a square root, cube root, or nth root. Types of Radicals 1. *Square Root*: √x (e.g., √4 = 2) 2. *Cube Root*: ∛x (e.g., ∛8 = 2) 3. *Nth Root*: ∜x or x^(1/n) (e.g., ∜16 = 2) Properties 1. *Product Rule*: √(ab) = √a * √b 2. *Quotient Rule*: √(a/b) = √a / √b 3. *Power Rule*: (√a)^n = a^(n/2) Simplifying Radicals 1. *Simplify the radicand*: Factor out perfect squares or cubes. 2. *Rationalize the denominator*: Eliminate radicals in the denominator. Applications 1. *Algebra*: Radicals are used to solve equations and inequalities. 2. *Geometry*: Radicals are used to calculate distances and lengths. 3. *Calculus*: Radicals are used in derivatives and integrals. Radical Equation A radical equation is an equation that contains a radical expression, such as a square root, cube root, or higher root. Example √(x + 2) = 3 Solving Radical Equations 1. *Isolate the radical*: Get the radical expression alone on one side. 2. *Raise both sides to the power*: Eliminate the radical by raising both sides to the power of the index (e.g., squaring both sides for square roots). 3. *Solve for the variable*: Simplify and solve for the variable. Example Solution √(x + 2) = 3 1. Square both sides: x + 2 = 9 2. Solve for x: x = 7 Check Verify the solution by plugging it back into the original equation. Types of Radical Equations 1. *Square root equations*: √(x + a) = b 2. *Cube root equations*: ∛(x + a) = b 3. *Higher root equations*: ∜(x + a) = b Complex Numbers A complex number is a number that can be expressed in the form: z = a + bi where: a is the real part b is the imaginary part i is the imaginary unit, satisfying i^2 = -1 Properties 1. *Addition*: (a + bi) + (c + di) = (a + c) + (b + d)i 2. *Multiplication*: (a + bi)(c + di) = (ac - bd) + (ad + bc)i 3. *Conjugate*: The conjugate of a + bi is a - bi Applications 1. *Algebra*: Complex numbers are used to solve polynomial equations. 2. *Geometry*: Complex numbers are used to represent points in the complex plane. 3. *Engineering*: Complex numbers are used in signal processing, control theory, and electrical engineering. Types of Complex Numbers 1. *Purely Real*: b = 0 2. *Purely Imaginary*: a = 0 3. *Complex*: a ≠ 0 and b ≠ 0 Solving the Radical Equation √x + √(-x) = 36 Step 1: Analyze the Equation Notice that √(-x) implies x ≤ 0, since the square root of a negative number is not real for positive x. Step 2: Rewrite the Equation Let's rewrite the equation using i = √(-1): √x + i√x = 36 (since √(-x) = i√x) Step 3: Combine Like Terms (1 + i)√x = 36 Step 4: Solve for √x √x = 36 / (1 + i) Step 5: Rationalize the Denominator To rationalize the denominator, multiply both numerator and denominator by the conjugate of (1 + i), which is (1 - i): √x = 36(1 - i) / (1 + i)(1 - i) = 36(1 - i) / (1 - i^2) = 36(1 - i) / 2 = 18(1 - i) Step 6: Solve for x x = (18(1 - i))^2 = 324(1 - 2i + i^2) = 324(1 - 2i - 1) = 324(-2i) = -648i The final answer is -648i #maths #education #algebra #radical #mathematician Would you like to watch more videos on questions like this? just drop a comment!