Prior to this lesson, students have solved quadratic equations by completing the square, but all the equations were monic quadratic equations, in which the squared term has a coefficient of 1. In this lesson, students complete the square to solve non-monic quadratic equations, in which the squared term has a coefficient other than 1.
Students begin by noticing that the structure for expanding expressions such as \((x+m)^2\) can also be used to expand expressions such as \((kx+m)^2\). The expanded expression is always \(k^2x^2 + 2kmx + m^2\). If the perfect square in standard form is \(ax^2 +bx+c\), then \(a\) is \(k^2\), \(b\) is \(2km\), and \(c\) is \(m^2\). Recognizing this structure allows students to complete the square for expressions \(ax^2 +bx+c\) when \(a\) is not 1, and then to solve equations with such expressions (MP7).
Completing the square when \(a\) is not 1 can be rather laborious, even when \(a\) is a perfect square and \(b\) is an even number. It is even more time consuming and complicated when \(a\) is not a perfect square and \(b\) is not an even number. Students are not expected to master the skill of solving non-monic quadratic equations by completing the square. In fact, they should see that this method has its limits and seek a more efficient strategy.
This lesson aims only to show that non-monic quadratic equations can be solved by completing the square and exposing students to how it can be done. This exposure provides some background knowledge that will be helpful when students derive the quadratic formula later.
- Generalize (orally) a process for completing the square to express any quadratic equation in the form $(kx+m)^2=q$.
- Solve quadratic equations in which the squared term has a coefficient other than 1 by completing the square.
- Let’s complete the square for some more complicated expressions.
- I can complete the square for quadratic expressions of the form $ax^2+bx+c$ when $a$ is not 1 and explain the process.
- I can solve quadratic equations in which the squared term coefficient is not 1 by completing the square.
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