At the end of this lesson, students will be able to:
- Complete the square of a quadratic expression.
- Solve quadratic equations by completing the square.
- Solve quadratic equations in standard form.
- Graph quadratic equations in vertex form.
- Solve real-world problems using functions by completing the square.
Terms introduced in this lesson:
completing the square
perfect square trinomial
quadratic equations in standard form
quadratic equations in vertex form
parabola turns up, turns down
Teaching Strategies and Tips
Use Example 1 to introduce completing the square.
- Remind students of binomial expansions to help them understand how a constant term turns the expression into a perfect square trinomial:
- Point out that the leading coefficient is .
- In Example 3, . Students learn to factor from the whole expression before completing the square, whether or not the terms are multiples of . Complete the square of the resulting expression in parentheses.
Give completing the square geometrical meaning.
- Use squares and rectangles for each term of the expression.
- See paragraph preceding Example 4.
- Have students make up a quadratic expression and ask them to complete the square in the geometrical interpretation as an assignment.
There are several reasons to have students learn completing the square:
- It is used to derive the quadratic formula.
- Quadratic equations can be rewritten in vertex form.
- Equations of circles can be rewritten in graphing form.
- Necessary in calculus.
Emphasize that completing the square finds roots
- Regardless of whether the roots are integers, rational or irrational numbers.
- Without having to list all the cases, unlike factoring.
Solve the following quadratic equation:
Solution: Although , and therefore and ; we show that completing the square results in the same solutions.
Divide by the leading coefficient; this results in an equation with .
Add the constant to both sides of the equation:
Factor the perfect square trinomial and simplify.
Take the square root of both sides:
Answer: and as expected.
Use Examples 7-9 to show how completing the square helps in graphing quadratic functions.
Dividing by the leading coefficient in Review Questions 7, 8, and 14 results in a fractional coefficient. Remind students that dividing by is equivalent to multiplying by .
Solve the quadratic equation by completing the square:
Hint: Rewrite the equation:
And divide by the leading coefficient:
To find the constant that must be added to both sides of the equation, multiply by and then square:
General Tip: Remind students to rewrite the vertex form of a quadratic equation with minus signs to find and correctly. See Examples 7-9.
Find the vertex of the parabola with equation:
Solution: Rewrite the equation with minus signs:
The vertex is at , where and .
General Tip: Remind students that the leading coefficient must be before completing the square. They will need to divide or factor to accomplish this.
General Tip: Suggest that students rewrite equations in standard form before completing the square.
In Examples 10 and 11 and Review Questions 25 and 26, have students round in the last step.