Overview
This lecture explains perfect squares and demonstrates how to solve quadratic equations by extracting square roots, both for perfect and non-perfect square cases.
Perfect Squares
- Perfect squares are numbers like 1, 4, 9, 16, 25, 36, etc.
- To extract the square root of a perfect square, take the principal square root and remember to include both positive and negative solutions.
Solving Quadratic Equations by Extracting Square Roots
- If ( x^2 = k ), then ( x = \pm\sqrt{k} ).
- For ( x^2 = 49 ), ( x = \pm 7 ).
- For ( x^2 = 169 ), ( x = \pm 13 ).
Non-Perfect Square Quadratic Equations
- If ( x^2 = n ) and ( n ) is not a perfect square, factor ( n ) to extract the square root of any perfect square within.
- For ( x^2 = 75 ): ( 75 = 25 \times 3 ), so ( x = \pm 5\sqrt{3} ).
- For ( x^2 = 80 ): ( 80 = 16 \times 5 ), so ( x = \pm 4\sqrt{5} ).
Quadratic Equations Involving Binomials
- For equations like ( 2(x-5)^2 = 32 ), first divide by 2, then extract the square root.
- ( (x-5)^2 = 16 ) leads to ( x - 5 = \pm4 ), so ( x = 9 ) or ( x = 1 ).
- Always write two equations: one for the positive root, one for the negative.
More Complex Examples
- For ( 3(4x-1)^2 - 1 = 11 ), isolate the squared term, divide, extract the square root, and solve for ( x ).
- Solutions may include fractions or surds (square roots of non-perfect squares).
Key Terms & Definitions
- Perfect Square — A number that is the square of an integer.
- Quadratic Equation — An equation of the form ( ax^2 + bx + c = 0 ).
- Extracting Square Roots — Solving for ( x ) by taking square roots of both sides of an equation.
- Surd — An irrational root, expressed with a square root sign (e.g., ( \sqrt{3} ), ( 3\sqrt{2} )).
Action Items / Next Steps
- Practice extracting square roots from quadratics with both perfect and non-perfect squares.
- Try to solve similar equations with binomials inside the square term.
- Review factoring techniques to simplify non-perfect squares.