COMPLETE THE SQUARE (+ leading coefficient "a" not 1) » algebraic & geometric methods | Math Hacks

Brett shows you how to complete the square to change a quadratic into vertex form using both the geometric area model and algebraic methods, including an example of completing the square when the leading coefficient of your quadratic is other than 1.

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SKIP AHEAD:
Example 1: Brett demonstrates how to complete the square with the quadratic: y = x^2 + 6x + 2 using the geometric method (1:00) and algebraic method (6:52).

Example 2: Brett demonstrates how to complete the square with the LEADING COEFFICIENT quadratic: y =2x^2 + 8x + 11 using the geometric method (11:16) and the algebraic method (15:23).

INSTRUCTIONS:
STEPS for completing the square GEOMETRICALLY (when leading coefficient a=1):
1. Draw a square sectioned into a medium sub-square, small sub-square, and two equal rectangles.
2. Assign the x^2 term to the area of the larger sub-square. The side lengths of the sub-square will by x.
3. Divvy up the x-term equally to the area of the two rectangles inside your square, so that half the x-term is assigned to each rectangle. Compute the missing side lengths of the rectangle.
4. Compute the area of the smaller sub-square by multiplying its side lengths together.
5. Write out the equation for the area of the large square using the outer side lengths and set it equal to the sum of the areas of the four inside squares and rectangles.
6. Adjust the equation by adding or subtracting a value so that the sum of areas equations matches the original equation. Make sure to add/subtract the value from both sides of the equation. The equation with the binomial squared term is the quadratic with its square completed.

STEPS for completing the square GEOMETRICALLY w/ a leading coefficient:

Follow the above steps, but begin by factoring out the leading coefficient from all three terms, then use the remaining equation to complete the square. After you finish the above steps, multiply the factored out value back through the equation.

STEPS for completing the square ALGEBRAICALLY (when leading coefficient a=1):

1. Rewrite your equation with parenthesis around the x-squared and x terms with extra space inside the parenthesis. Leave the constant outside of the parenthesis.
2. Find the coefficient on the x term, divide it by 2 and square it. Add this value to the blank space inside the parenthesis.
3. Subtract the value found in step 2 from the constant on the outside of the parenthesis.
4. Factor the trinomial inside of the parenthesis (this should factor into a binomial squared).

STEPS for completing the square ALGEBRAICALLY w/ leading coefficient:

Follow the above steps but after step 1 factor out the leading coefficient from the x^2 and x terms. Follow step 2 as stated above. Then on step 3 make sure to subtract the value added in step 2 TIMES the coefficient factored out. This ensures that the equation stays unchanged.

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