Mastery Check Solutions

Show What You Know

Follow each direction below using the equation $a x^{2} + b x + c = 0$ to show the work.

Note

This Mastery Check is critical to student understanding and success with this lesson and the next one. Be sure that your student completes this carefully.

Direction	Show Work Using $a x^{2} + b x + c = 0$
Subtract c from both sides of the equation.	$a x^{2} + b x + c = 0 - c - c$
Divide all terms by the coefficient a.	$\begin{array}{rcl} \frac{a x^{2}}{a} + \frac{b x}{a} & = & \frac{- c}{a} \\ x^{2} + \frac{b}{a} x & = & \frac{- c}{a} \end{array}$
Add b to the second power divided by 4 times a to the second power to both sides.	$x^{2} + \frac{b}{a} x + \frac{b^{2}}{4 a^{2}} = \frac{- c}{a} + \frac{b^{2}}{4 a^{2}}$
Write the left side of the equation as a binomial squared.	$\begin{array}{rcl} (x + \frac{b}{2 a}) (x + \frac{b}{2 a}) & = & \frac{- c}{a} + \frac{b^{2}}{4 a^{2}} \end{array}$
Write the right side of the equation as one fraction with a common denominator, $4 a^{2} .$	$\begin{array}{rcl} = & \frac{- c}{a} (\frac{4 a}{4 a}) + \frac{b^{2}}{4 a^{2}} \\ {(x + \frac{b}{2 a})}^{2} & = & \frac{b^{2} - 4 a c}{4 a^{2}} \end{array}$
Take the square root of both sides of the equation.	$\begin{array}{rcl} \sqrt{{(x + \frac{b}{2 a})}^{2}} & = & \pm \sqrt{\frac{b^{2} - 4 a c}{4 a^{2}}} \\ x + \frac{b}{2 a} & = & \frac{\pm \sqrt{b^{2} - 4 a c}}{2 a} \end{array}$
Isolate the variable x.	$\begin{array}{rcl} x & = & - \frac{b}{2 a} \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a} \end{array}$
Write the right-side of the equation as one fraction.	$\begin{array}{rcl} x & = & \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{array}$

Note

1–2)

Q: Would your answer be the same if steps 1 and 2 were reversed? Explain.

A: Yes, the value will remain the same.

Remember that every term on both sides must be divided by a.
If you need the verbal description as an expression, it is $\frac{b^{2}}{4 a^{2}} .$
You can also factor the left side as a perfect square trinomial.
Remember to include ± when taking the square root of the right side.
To write the fraction as one equation, the numerators will be combined since there is already an LCD.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

- Complete the square when $a = 1 .$
- Complete the square when $a \neq 0, 1 .$