Q&A for How to Complete the Square

It does seem strange and arbitrary, but there is a reason for it. The power move is taking the square root of both sides, but you can't simplify the square root of most polynomials. The step you ask about is a setup move to make the power move work. If I have, for example, x^2 + 4x = 5, and take the square root of both sides, nothing happens, it just makes a mess. But if I add 4 to both sides first and take the square root of both sides of x^2 + 4x + 4 = 9, it simplifies to |x+2| = 3 and the quadratic equation is reduced to a linear equation.

The value of x doesn't matter. The process remains as shown above.

Both sides of that equation are being divided by 3 (to get rid of the coefficient of the first term). Dividing the second term (11/3) by 3 gives us 11/9.

When no coefficient is shown, you can consider the coefficient to be 1.

Use a calculator to find the square root of 11,111. It's about 105.4. Take the next higher whole number (106), square it (11,236), then subtract 11,111 to find your answer.

Notice that 4 and 49 are both perfect squares. So the factors you'll use for 4 will be either 2 and 2 or -2 and -2. The factors for 49 will be either 7 and 7 or -7 and -7. You'll have to multiply the 2's and the 7's together and add them to arrive at -28. Notice that 2 x -7 equals -14, and two of those would add to make -28. So the factors would be (2x - 7) and (2x - 7) or (2x - 7)². Note that you could change the signs in your factors and still get the same result: (7 - 2x)(7 - 2x) also equals 4x² - 28x + 49. So you could write the final answer as +/-(2x-7)².

Solve it graphically. Plot the graph of that function, and note the coordinates of the vertex (which is a minimum). The y coordinate of the vertex is the lowest value of the function, and the x coordinate of the vertex is the x value at which this lowest y-value occurs.

It's because we want to make a blank space to complete the square.

First take 8 (the coefficient of r), divide it by 2, which is 4. Then square 4, which is 16. Now add 16 to both sides of the equation: r² - 8r + 16 = -8 + 16 = 8. Then (r - 4)² = 8. Now take the square root of both sides: (r - 4) = ±√8. Then r = 4 ±√8. (Leave the answer in that form, or if you prefer, evaluate √8 and then add it to and subtract it from 4.)