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Completing the Square

Completing the Square. To review how to solve quadratic equations in vertex form To solve quadratic equations in general form by completing the square. Recall that rectangle diagrams help you factor some quadratic expressions.

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Completing the Square

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  1. Completing the Square To review how to solve quadratic equations in vertex form To solve quadratic equations in general form by completing the square

  2. Recall that rectangle diagrams help you factor some quadratic expressions.

  3. How do you find the roots of an equation such as 0=x2+x-1? It is not a perfect-square trinomial, nor is it easily factorable. For these equations you can use a method called completing the square.

  4. Searching for Solutions • To understand how to complete the square with quadratic equations, you’ll first work with rectangle diagrams. • Complete each rectangle diagram so that it is a square. How do you know which number to place in the lower-right corner?

  5. 4 9 6.25 Searching for Solutions • For each diagram, write an equation in the formx2+bx+c=(x+h)2. • On which side of the equation can you isolate x by undoing the order of operations? 9

  6. 4 9 6.25 Searching for Solutions • Suppose the area of each diagram is 100 square units. • For each square, write an equation that you can solve for x by undoing the order of operations. • Solve each equation symbolically. You will get two values for x.

  7. The solutions for x in the equations from the last slide are rational numbers. This means you could have factored the equations with rational numbers. However, the method of completing the square works for other numbers as well. Next you’ll consider the solution of an equation that you cannot factor with rational numbers.

  8. Consider the equation x2+6x-1=0. Describe what’s happening in each stage of the solution process.

  9. Use your calculator to find decimal approximations for and . • Then enter the equation y=x2+6x-1 into Y1. Use a calculator graph or table to check that your answers are the x-intercepts of the equation.

  10. Repeat the solution stages on to find the solutions to x2+8x-5=0.

  11. The key to solving by completing the square is to express one side of the equation as a perfect-square trinomial. • In the investigation the equations are in the form y=1x2+bx+c. Note that the coefficient of x2, called the leading coefficient, is 1. However, there are other perfect-square trinomials. • An example of perfect-square trinomial with a different leading coefficient is shown at right.

  12. Example A • Solve the equation 3x2+18x-8=22 by completing the square. • First, transform the equation so that you can write the left side as a perfect-square trinomial in the form x2+2hx+h2.

  13. Now you need to decide what number to add to both sides to get a perfect-square trinomial on the left side. • Use a rectangle diagram to make a square. When you decide what number to add, you must add it to both sides to balance the equation.

  14. The two solutions are , or approximately 1.36, and , or approximately -7.36.

  15. Example B • Find the vertex form of the equation y =2x2+12x+21. • Then identify the vertex and any x-intercepts of the parabola. • To convert y =2x2+12x+21 to the form y=a(x-h)2+k, complete the square. • Now you can complete the square on the expression inside the parentheses.

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