Solving Quadratic Equations Involving Radicals

Solving Quadratic Equations Involving Radicals

In mathematics, radical equations often involve quadratic functions, which require special handling. These equations contain variables raised to a power, such as x^2 or 2x, and involve operations with radical signs, such as the square root symbol √. To solve these types of equations, we need to follow a particular process.

Steps to Solve Radical Equations

Here's a step-by-step guide on how to solve a quadratic equation that involves radicals:

1. Isolate the radical: The key is to ensure that the radical expression appears alone on one side of the equation. This might involve moving constants to the other side or combining like terms.

2. Square both sides: Once the radical is isolated, we square both sides of the equation. Note that when squaring both sides, we also need to square the constant coefficient of the radical expression.

3. Simplify: After squaring the equation, we try to simplify it as much as possible. This may involve bringing down exponents and applying the distributive property where necessary.

4. Solve the new equation: With the radical eliminated, we treat this as a regular quadratic equation, aiming to set it to zero. We factor or use the quadratic formula to find the solutions.

5. Check the answer: Finally, we check our answers by substituting them back into the original equation to confirm that they satisfy the condition.

When dealing with equations that have multiple radicals, simply repeat steps 1 to 4 until all radicals are removed.

Examples

Let's consider some examples to illustrate this process:

Example 1: ‣ Solve the equation (\sqrt{x} - 2 = 0\).

Step 1: Isolate the radical.

‣ Add 2 to both sides of the equation: \(\sqrt{x} = 2\).

Step 2: Raise both sides to the power of 2 (since the exponent of the square root is 2):

‣ \(x = 4\).

Now we have found the first solution, (x = 4). However, since the radical exists on one side of the equation, we need to apply the reverse operation (take the square root) to the new equation and get:

‣ \(x = 4 ^ { 2 } - 2\) = \(16 - 2\) = \(14\).

So, another valid solution is (x = 14).

Example 2: ‣ Solve the equation (\left(x + 3\right)^2 - 10 = 0\).

Step 1: Isolate the radical.

‣ Subtract 3 from both sides of the equation: \(\left(x + 3\right)^2 = 10\).

Step 2: Raise both sides to the power of 2:

‣ \(x ^ { 2 } + 6 x + 9 = 10\).

Now we have a quadratic equation. Let's solve it:

‣ \(x ^ { 2 } + 6 x - 1 = 0\).

We can easily factor this quadratic equation: \((x + 3)(x - 1) = 0\). Setting each factor equal to 0, we obtain the solutions x = -3 and x = 1.