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Some equations can be transformed or manipulated into a quadratic equation, which makes them easier to solve. Equations that can be reduced to quadratic form often involve variables raised to powers or contain multiple terms.

Types of Equations

Equations that are reducible to quadratic form include:

• Equations involving radicals: Equations with square roots or other radicals can often be squared to eliminate the radical and form a quadratic equation.
• Equations with rational expressions: Equations containing rational expressions can sometimes be transformed into quadratic equations by clearing denominators.
• Equations with variable substitutions: Substituting a new variable can sometimes transform a given equation into a quadratic equation.
• Equations involving trigonometric functions: Trigonometric identities or trigonometric substitutions can sometimes reduce trigonometric equations to quadratic form.

Example

Let's consider the equation $$\frac{1}{x}-\frac{1}{y}=3$$. We can rewrite it as:

Now, let's make a substitution, $$z=xy$$. This transforms the equation into a quadratic equation:

This equation is now in quadratic form, and we can solve it using quadratic equation-solving techniques.

Understanding how to manipulate equations into quadratic form can simplify the solving process and make it more manageable.