Completing The Square

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A technique named "completing the square" is used to rewrite a quadratic equation in what is known as the "standard form." It involves adding and subtracting the equation's variables in order to produce a perfect square trinomial on the left-hand side of the equation. 

Here is a step-by-step instruction sheet for finishing the square:

  1. Write the quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.
  2. Find out the value of a. If a is not 1, divide the entire equation by a to make it equal to 1. This will make sure that the coefficient of the x^2 term is 1.
  3. Rewrite the equation as x^2 + (b/a)x + (c/a) = 0.
  4. Add and subtract (b/2a)^2 to the equation. This will make a perfect square trinomial on the left-hand side of the equation.
  5. Make the equation simple. If you have added and subtracted the same value, the equation will simplify to (x + (b/2a))^2 = (b/2a)^2 - (c/a).
  6. Take the square root of both sides to solve the equation. The values of x that make the original equation true are indeed the solutions to the equation.

When the quadratic formula is not able to obtain an answer, completing the square is a useful method for attempting to solve quadratic equations. It can be used to create quadratic functions and calculate a quadratic function's maximum or minimum value.

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