Extracting Square Roots Practice Problems & Solutions

Master quadratic equations by extracting square roots with step-by-step practice problems. Learn to solve x² - c = 0 equations efficiently and avoid common mistakes.

📚Master Square Root Extraction with Interactive Practice

Solve quadratic equations of the form x² - c = 0 using square root extraction
Master the three-step process: isolate x², take square roots, write solutions
Handle coefficients greater than 1 by dividing to isolate x² completely
Recognize when equations have no solution due to negative square roots
Work with fractional and radical solutions confidently
Apply the ± symbol correctly when extracting square roots

Understanding Extracting Square Roots

Complete explanation with examples

Solution by extracting a root

Solving a quadratic equation with one variable $X^2-c=0$ (where $b=0$ ) by calculating the square root:

First step

Moving terms and isolating $X^2$ .

Second stage

Take the square root of both sides. Don't forget to insert $\pm$ before the square root of the free term.

Third stage

Writing solutions in an organized manner or writing "no solution" in case of a root of a negative number.

Detailed explanation

Practice Extracting Square Roots

Test your knowledge with 15 quizzes

Solve the following equation

$$ x^2-25=0 $$

Examples with solutions for Extracting Square Roots

Step-by-step solutions included

Exercise #1

Solve the following exercise:

$2x^2-8=x^2+4$

Step-by-Step Solution

First, we move the terms to one side equal to 0.

$2x^2-x^2-8-4=0$

We simplify :

$x^2-12=0$

We use the shortcut multiplication formula to solve:

$x^2-(\sqrt{12})^2=0$

$(x-\sqrt{12})(x+\sqrt{12})=0$

$x=\pm\sqrt{12}$

Answer:

$±\sqrt{12}$

Video Solution

Exercise #2

Solve the following exercise:

$x^2-20=5$

Step-by-Step Solution

To solve this quadratic equation $x^2 - 20 = 5$ , we will follow these steps:

Step 1: First, add 20 to both sides of the equation to isolate the $x^2$ term:

$x^2 - 20 + 20 = 5 + 20$

This simplifies to:

$x^2 = 25$

Step 2: Next, take the square root of both sides to solve for $x$ :

$x = \pm \sqrt{25}$

$x = \pm 5$

Therefore, the solutions to the equation are:

$x = 5$ and $x = -5$

Thus, the correct answer choice is:

$\pm 5$ from the provided options.

The correct solution to the problem is $\pm 5$ .

Answer:

±5

Video Solution

Exercise #3

Solve for X:

$x\cdot x=49$

Step-by-Step Solution

We first rearrange and then set the equations to equal zero.

$x^2-49=0$

$x^2-7^2=0$

We use the abbreviated multiplication formula:

$(x-7)(x+7)=0$

$x=\pm7$

Answer:

±7

Video Solution

Exercise #4

Solve the following:

$x^2+x^2-3=x^2+6$

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Simplify the given equation.
Solve for $x^2$ and find $x$ .

First, let's simplify the equation:

$x^2 + x^2 - 3 = x^2 + 6$ .

Combine like terms on the left side:

$2x^2 - 3 = x^2 + 6$ .

Subtract $x^2$ from both sides to isolate one of the $x$ terms:

$2x^2 - x^2 - 3 = 6$ .

This simplifies to:

$x^2 - 3 = 6$ .

Next, add 3 to both sides to solve for $x^2$ :

$x^2 = 9$ .

To find $x$ , take the square root of both sides:

$x = \pm\sqrt{9}$ .

This results in:

$x = \pm3$ .

Therefore, the solution to the problem is $\pm3$ .

Answer:

±3

Video Solution

Exercise #5

Solve the following equation:

^{$4x^2+8+2x=x+12+x$}

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Simplify both sides of the equation.
Step 2: Rearrange terms to form a quadratic equation.
Step 3: Solve the quadratic equation using an appropriate method, such as factoring or applying the quadratic formula.

Now, let's work through each step:

Step 1: Simplify both sides of the equation:

4x^2 + 8 + 2x = x + 12 + x

The right-hand side can be simplified:

x + x + 12 = 2x + 12

This yields the equation:

4x^2 + 8 + 2x = 2x + 12

Step 2: Rearrange the terms to form a quadratic equation. Subtract $2x + 12$ from both sides:

4x^2 + 8 + 2x - 2x - 12 = 0

Combining like terms gives:

4x^2 - 4 = 0

Step 3: Solve the resulting quadratic equation:

First, we add 4 to both sides:

4x^2 = 4

Next, divide both sides by 4:

x^2 = 1

Now, apply the square root to both sides:

x = \pm 1

Therefore, the solutions to the quadratic equation are

$x = \pm 1$ .

The correct answer is choice 2: ±1.

Answer:

±1

Video Solution

Frequently Asked Questions

Everything you need to know about Extracting Square Roots

What is the square root extraction method for quadratic equations?

The square root extraction method is used to solve quadratic equations of the form x² - c = 0 (where b = 0). It involves three steps: isolate x² on one side, take the square root of both sides with ± symbol, and write the two solutions clearly.

When can I use square root extraction instead of the quadratic formula?

Use square root extraction when your quadratic equation has no middle term (b = 0) and looks like ax² + c = 0. This method is much faster than the quadratic formula for these specific equations.

Why do I need to use ± when extracting square roots?

The ± symbol is essential because every positive number has two square roots: one positive and one negative. For example, both 4 and -4 when squared equal 16, so x = ±4 gives you both solutions.

What happens if I get a negative number under the square root?

If you need to find the square root of a negative number, the equation has no real solution. You cannot find a real number that when squared gives a negative result, so write 'no solution' as your answer.

How do I handle coefficients in front of x² when extracting square roots?

First divide both sides by the coefficient to completely isolate x². For example, with 3x² = 9, divide both sides by 3 to get x² = 3, then take the square root: x = ±√3.

Should I simplify square roots in my final answer?

Yes, always simplify square roots when possible. For example, √16 should be written as 4, not left as √16. However, if the square root cannot be simplified to a whole number, it's acceptable to leave it in radical form.

What are the most common mistakes when extracting square roots?

Common mistakes include: 1) Forgetting the ± symbol, 2) Not completely isolating x² before taking the square root, 3) Trying to find the square root of a negative number instead of writing 'no solution', and 4) Not simplifying final answers when possible.

Can I use square root extraction for equations like x² + 4 = 0?

Yes, you can apply the method, but this specific equation has no real solution. After isolating x², you get x² = -4, and since you cannot take the square root of a negative number, the answer is 'no solution'.

Continue Your Math Journey

Master Extracting Square Roots first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Completing the square in a quadratic equation

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Applying the formula Factoring Out the Greatest Common Factor (GCF) Identify the greater value Solving the equation Using short multiplication formulas With fractions

Extracting Square Roots Practice Problems & Solutions

Understanding Extracting Square Roots

Solution by extracting a root

First step

Second stage

Third stage

Practice Extracting Square Roots

Examples with solutions for Extracting Square Roots

Frequently Asked Questions

What is the square root extraction method for quadratic equations?

When can I use square root extraction instead of the quadratic formula?

Why do I need to use ± when extracting square roots?

What happens if I get a negative number under the square root?

How do I handle coefficients in front of x² when extracting square roots?

Should I simplify square roots in my final answer?

What are the most common mistakes when extracting square roots?

Can I use square root extraction for equations like x² + 4 = 0?

More Extracting Square Roots Questions