Solve each quadratic equation using the square root property. See Example 6. x² - 27 = 0

Verified step by step guidance

1. Start with the given quadratic equation: \(x^{2} - 27 = 0\).
2. Isolate the squared term by adding 27 to both sides: \(x^{2} = 27\).
3. Apply the square root property, which states that if \(x^{2} = a\), then \(x = \pm \sqrt{a}\).
4. Take the square root of both sides: \(x = \pm \sqrt{27}\).
5. Simplify the square root if possible by factoring out perfect squares: \(\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\), so \(x = \pm 3\sqrt{3}\).

Key Concepts

Square Root Property

The square root property states that if x² = k, then x = ±√k. This method is used to solve quadratic equations that can be written in the form x² = constant by isolating x² and then taking the square root of both sides, considering both positive and negative roots.

Isolating the Variable Term

Before applying the square root property, the quadratic equation must be rearranged so that the x² term is isolated on one side of the equation. This often involves adding or subtracting constants to both sides to simplify the equation into the form x² = number.

Handling Negative Values Under the Square Root

If the constant on the right side of the equation is negative, the solution involves imaginary numbers because the square root of a negative number is not real. In such cases, the solutions are expressed using the imaginary unit i, where i² = -1.