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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