Extracting square roots in a quadratic equation is a method of solving for the variable when the equation can be manipulated into the form (ax + b)² = c. It involves isolating the squared term and then applying the square root property to solve for the unknown.
Understanding the Process
Extracting square roots is a straightforward technique for solving quadratic equations that are in a specific form. Here's a breakdown:
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Isolate the Squared Term: The first step is to manipulate the equation so that the squared term (e.g., (x + 3)²) is by itself on one side of the equation.
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Apply the Square Root Property: The square root property states that if
u² = d, thenu = √doru = -√d. This is crucial because most numbers have two square roots: a positive and a negative one. -
Solve the Resulting Equations: After applying the square root property, you will have two separate linear equations. Solve each of these equations for the variable.
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Simplify: Simplify any radical expressions. If the denominator of a solution contains a radical, rationalize the denominator.
Example
Let's consider the quadratic equation:
(x + 2)² = 9
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Squared Term is already isolated: (x + 2)² is already by itself.
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Apply the Square Root Property:
x + 2 = √9 or x + 2 = -√9
x + 2 = 3 or x + 2 = -3
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Solve the equations:
x = 3 - 2 or x = -3 - 2
x = 1 or x = -5
Therefore, the solutions to the quadratic equation (x + 2)² = 9 are x = 1 and x = -5.
When to Use Extracting Square Roots
This method is most effective when:
- The quadratic equation is in the form (ax + b)² = c.
- There is no 'bx' term after expansion.
It is not ideal for quadratic equations in the standard form ax² + bx + c = 0 if 'b' is not zero and the equation isn't easily manipulated into the (ax + b)² = c form. In such cases, factoring, completing the square, or using the quadratic formula may be more appropriate.
