How to Complete the Square

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When you are unable to solve a quadratic equation of the form ax² +bx+c by factoring, then you can use the technique called completing the square. To complete the square means to create a polynomial with three terms (trinomial) that is a perfect square.

The Complete the Square Method

    Rewrite the quadratic expression ax² +bx+c in the form ax² +bx= -c by moving the constant term c to the right side of the equation.

    Take the equation in Step 1 and divide by the constant a if a≠ 1 to get x² + (b/a) x = -c/a.

    Divide (b/a) which is the x term coefficient by 2 and this becomes (b/2a) then square it (b/2a)².

    Add the (b/2a)² to both sides of the equation in Step 2: x² + (b/a) x + (b/2a)² = -c/a + (b/2a)².

    Write the left side of the equation in Step 4 as a perfect square: [x + (b/2a)]² = -c/a + (b/2a)².

Apply the Complete the Square Method

    Complete the square of the expression 4x²+16x-18. Note that a=4, b=16 c= -18.

    Move the constant c to the right side of the equation to get 4x²+16x= 18. Remember that when you move -18 to the right side of the equation it becomes positive.

    Divide both sides of the equation in Step 2 by 4: x²+ 4x= 18/4.

    Take ½ (4) which is the x term coefficient in Step 3 and square it to get (4/2)²=4.

    Add the 4 from Step 4 to both sides of the equation: in Step 3: x²+ 4x +4= 18/4 + 4. Change the 4 on the right side to the improper fraction 16/4 to add like denominators and rewrite the equation as x²+ 4x +4= 18/4 + 16/4= 34/4.

    Write the left side of the equation as (x+2)² which is a perfect square and you get that (x+2)²= 34/4.This is the answer.

    Tips

    • The additive inverse property states that a + (-a) =0. Be careful of the signs when you move the constant to the right side of the equation.

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