Solving Quadratic Equations Involving Radicals

Study Notes

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:

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.
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.
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.
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.
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.

Description

Learn how to solve quadratic equations that involve radical expressions step by step. Understand the process of isolating the radical, squaring both sides, simplifying, solving the new equation, and checking the answers. Practice with examples to reinforce your understanding.

Solving Quadratic Equations Involving Radicals

Choose a study mode

Podcast

Questions and Answers

What is the first step in solving a quadratic equation involving radicals?

What is the second step after isolating the radical in a radical equation involving quadratics?

What should be done after squaring both sides of the equation in a radical equation involving quadratics?

What do you do after simplifying the equation in a radical equation involving quadratics?

What is the final step in solving a quadratic equation involving radicals?

How do you solve the equation (`\sqrt{x} - 2 = 0`)?

What is the solution to the equation (`\left(x + 3\right)^2 - 10 = 0`)?

When dealing with equations that have multiple radicals, what should you do?

What is the importance of raising both sides to the power of 2 when dealing with square roots?

How do you find the solutions to a quadratic equation after isolating the radical?

Study Notes