Math Review of the Square Root Property

Overview

The square root property is an important shortcut to use when solving quadratic equations, or any equation that asks for square roots and powers of 2. It uses the definition of a square root to solve for a variable.

Definition

The definition of the square root of a positive real number in symbol form is if x² = a, then x = ±√a. For example, the square root of 64 is ±8, because 8∙8 is 64 and -8∙-8 is also 64. Suppose that y² = b. Using substitution, y = ±√b. The variable b can stand for a positive real number, a monomial, or even a polynomial.

Using the Square Root Property with Positive Real Numbers

Suppose that x² = 100. Then x will equal ±√100, or 10 or -10. Suppose that y² + 11 = 92. Then y² = 92 – 11. If y² = 81, then y will equal ±√81, or 9 or -9. Suppose that x² = 49/25. Then x will equal √(49/25) or ±7/5.

Figure 1: The square root property holds with positive real numbers and variables.

Using the Square Root Property with Monomials

The square root property can also be used with monomials. Suppose the equation is 36x⁴y² = z². Using the square root property, z equals √(36x⁴y²) or 6x²y, because the square root of 36 is 6, the square root of x⁴ is x² and the square root of y² is y. Similarly, if q² equals 20a⁴b² then q will equal 2 a²b√5. This is because the square root of 20 equals 2√5, the square root of a⁴ is a² and the square root of b² is b.

Figure 2: The process of using the square root property with a monomial.

Using the Square Root Property with Quadratic Equations

Using the square root property can be a shortcut to solving perfect square trinomials. Suppose the equation is x² + 6x + 9 = 36. In this case, both sides are perfect squares, because the perfect square trinomial is (x +3)² = 6². Therefore x + 3 = ±6. Solving for x + 3 = 6, x = 6 -3 or 3. Solving for x + 3 = -6, x = -6 – 3, or -9.

Figure 3: Using the square root property to solve a quadratic equation.

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