Which of the following quadratic equations can be solved by extracting square roots?

Question 3Answer

a.
x² - 25 = 0


b.
x² + 4x + 1= 0


c.
x² + 3x = 0


d.
4x² + 3x + 12 = 0

To determine which quadratic equations can be solved by extracting square roots, the key is identifying those that can be simplified into a form like \(x^2 = k\), where \(k\) is a constant, without the need for factoring or using the quadratic formula.

Let's analyze each equation:

**a. \( x^2 - 25 = 0 \)**  
This equation can be rewritten as \(x^2 = 25\), so you can solve by taking square roots.  
Thus, **this equation can be solved by extracting square roots**.

**b. \( x^2 + 4x + 1 = 0 \)**  
This is a full quadratic equation with a linear term (\(4x\)) and does not fit the form \(x^2 = k\). Solving it requires the quadratic formula or factoring.  
Thus, **this equation cannot be solved by extracting square roots**.

**c. \( x^2 + 3x = 0 \)**  
This equation can be factored as \(x(x + 3) = 0\), but it cannot be directly solved by extracting square roots because of the linear term. Factoring or the quadratic formula is needed.  
Thus, **this equation cannot be solved by extracting square roots**.

**d. \( 4x^2 + 3x + 12 = 0 \)**  
This is also a full quadratic equation that includes both a linear term and a constant. It cannot be reduced to the form \(x^2 = k\) for extracting square roots.  
Thus, **this equation cannot be solved by extracting square roots**.

### Answer: **a. \( x^2 - 25 = 0 \)** is the only one that can be solved by extracting square roots.

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Would you like any further details or clarifications?

Here are 5 related questions to further your understanding:

1. What is the general form of a quadratic equation?
2. How do you solve a quadratic equation using the quadratic formula?
3. What are the conditions for factoring a quadratic equation?
4. How can completing the square help in solving quadratic equations?
5. When can we use the method of extracting square roots for solving equations?

**Tip:** To solve by extracting square roots, the equation should have no linear term and should be reducible to the form \(x^2 = k\).

Which of the following quadratic equations can be solved by extracting square roots?

a. x² - 25 = 0
b. x² + 4x + 1 = 0
c. x² + 3x = 0
d. 4x² + 3x + 12 = 0

Learn how to determine which quadratic equations can be solved by extracting square roots. Explore step-by-step explanations for solving equations like x² - 25 = 0 and why other forms require different methods. Suitable for students in Grades 8-10.

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