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10.3: Quadratic Equations by Square Roots

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

At the end of this lesson, students will be able to:

• Solve quadratic equations involving perfect squares.
• Approximate solutions of quadratic equations.
• Solve real-world problems using quadratic functions and square roots.

Vocabulary

Terms introduced in this lesson:

positive, negative square root
approximate solutions
free fall

Teaching Strategies and Tips

Remind students that:

• x2=|x|\begin{align*}\sqrt{x^2}=|x|\end{align*} and is called the principal square root.
• |x|=k\begin{align*}|x|=k\end{align*} implies x=±k\begin{align*}x = \pm k\end{align*}.
• Conclude that x2=c\begin{align*}x^2=c\end{align*} has two solutions when c>0\begin{align*}c > 0\end{align*}. See Examples 1-3.
• Point out that square roots of negative numbers are not real numbers. Therefore, x2=c\begin{align*}x^2=c\end{align*} has no solutions when c<0\begin{align*}c < 0\end{align*}.

Specify in advance the number of decimal places required of students in Examples 6 and 7 and Review Questions 18-22.

Error Troubleshooting

In Example 2, show students that 169=43\begin{align*}\sqrt{\frac{16}{9}}=\frac{4}{3}\end{align*} and 181=19\begin{align*}\sqrt{\frac{1}{81}}=\frac{1}{9}\end{align*}. More square-roots of fractions can be found in Review Questions 4-6 and 8.

General Tip: Remind students after square-rooting both sides of an equation to include the ±\begin{align*}\pm\end{align*}.

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