Question

Solve the quadratic equation by extracting square roots. List both the exact answer and a decimal answer that has been rounded to two decimal places. $$ 3 x^{2}=36 $$

Short answer

The solutions to the equation are \(x = \sqrt{12}\) and \(x = -\sqrt{12}\), or when rounded off to two decimal places, \(x = 3.46\) and \(x = -3.46\).

Step-by-step solution

Isolate \(x^{2}\)

Divide both sides of the equation \(3x^{2} = 36\) by 3 to get \(x^{2}\) alone. This gives us \(x^{2} = 12\).

Extract square roots

Take the square root of both sides of the equation \(x^{2} = 12\). Because both a positive and negative number squared will result in a positive number, the answer is both \(\sqrt{12}\) and \(-\sqrt{12}\).

Simplify Square Roots and Round to Two Decimal Places

No further simplification can be done for \(\sqrt{12}\) so we convert it to decimal form which is around 3.46. The answers will be \(3.46\) and \(-3.46\).

Key concepts

Solving by Extracting Square Roots

When you encounter a quadratic equation of the form \( ax^2 = c \), the simplest method to find the value of \( x \) is often extracting square roots. This technique involves the following steps which make it straightforward:

• Isolate the squared term: Start by dividing both sides of the equation by the coefficient of \( x^2 \) to get \( x^2 \) alone on one side.
• Take the square root: Once you've isolated \( x^2 \), take the square root of both sides of the equation. Remember, you must include both the positive and negative solutions because squaring either of these will return you to the positive value on the left side.

For example, with the equation \( 3x^2 = 36 \), you start by dividing both sides by 3 to get \( x^2 = 12 \). Then, you take the square root of both sides which results in the solutions \( x = \sqrt{12} \) and \( x = -\sqrt{12} \). This approach effectively uses the property that both positive and negative roots exist for any squared variable.

Exact and Decimal Answers

After extracting the square roots, you will often have an exact answer expressed with square root symbols. However, to apply these solutions practically, we sometimes need their decimal approximations.Here’s how you can address both exact and decimal answers:

• Exact Answer: Maintain the square root form for maximum precision. For the equation \( x^2 = 12 \), the exact solutions are \( x = \sqrt{12} \) and \( x = -\sqrt{12} \).
• Decimal Answer: Use a calculator to find a decimal form for the solution. After computing, you should round the number to two decimal places for ease of use in real-world applications. In this example, \( \sqrt{12} \approx 3.46 \), and thus the decimal solutions are \( x \approx 3.46 \) and \( x \approx -3.46 \).

This process not only solidifies your understanding but also prepares you for applying solutions in both theoretical and practical scenarios.

Simplifying Square Roots

Simplifying square roots is an essential skill, especially when working with quadratic equations, as it helps in reducing the expression to a more manageable form.To simplify a square root, follow these steps:

• Identify perfect squares: Look for perfect square factors within the square root. For example, for \( \sqrt{12} \), you can factor 12 as 4 (a perfect square) times 3.
• Separate and simplify: Use the property \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \). For \( \sqrt{12} \), this would become \( \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \).
• Write the simplified form: Once simplified, write down the expression in its simplest square root form, which can prevent further calculation errors.

For the problem \( x^2 = 12 \), while simplification reveals \( \sqrt{12} = 2\sqrt{3} \), it does not further influence the decimal conversion as the numeric result remains \( \approx 3.46 \). Simplified expressions, however, can often make other mathematical operations and simplifications much clearer.