Lesson 11

In each equation, what expression could be substituted for \(a\) so the equation is true for all values of \(x\)?

1. \(x^2 = a^2\)
2. \((3x)^2=a^2\)
3. \(a^2=7x \boldcdot 7x\)
4. \(25x^2=a^2\)
5. \(a^2=\frac14 x^2\)
6. \(a^2=(x+1)^2\)
7. \((2x-9)(2x-9)=a^2\)

1. Each expression is written as the product of factors. Write an equivalent expression in standard form.

   1. \((3x)^2\)
   2. \(7x \boldcdot 7x\)
   3. \((x+4)(x+4)\)
   4. \((x+1)^2\)
   5. \((x-7)^2\)
   6. \((x+n)^2\)

2. Why do you think the following expressions can be described as perfect squares?

   \(x^2+6x+9 \qquad x^2-16x+64 \qquad x^2+\frac13 x + \frac{1}{36}\)

Han and Jada solved the same equation with different methods. Here they are:

Work with a partner to solve these equations. For each equation, one partner solves with Han’s method, and the other partner solves with Jada’s method. Make sure both partners get the same solutions to the same equation. If not, work together to find your mistakes.

 

These are some examples of perfect squares:

• 49, because 49 is \(7 \boldcdot 7\) or \(7^2\).
• \(81a^2\), because it is equivalent to \((9a)\boldcdot(9a)\) or \((9a)^2\).
• \((x+5)^2\), because it is equivalent to \((x+5)(x+5)\).
• \(x^2-12x+36\), because it is equivalent to \((x-6)^2\) or \((x-6)(x-6)\).

A perfect square is an expression that is something times itself. Usually we are interested in situations in which the something is a rational number or an expression with rational coefficients.

When expressions that are perfect squares are written in factored form and standard form, there is a predictable pattern.

• \((x+5)(x+5)\) is equivalent to \(x^2+10x+25\).
• \((x-6)^2\) is equivalent to \(x^2-12x+36\).
• \((x-9)^2\) is equivalent to \(x^2-18x+81\).

In general, \((x+n)^2\) is equivalent to \(x^2+(2n)x+n^2\).

Quadratic equations that are in the form \(\text {a perfect square} = \text {a perfect square}\) can be solved in a straightforward manner. Here is an example:

\(\displaystyle \begin {align} x^2 - 18x + 81 &= 25 \\(x-9)(x-9) &=25\\ (x - 9)^2 &= 25 \end {align}\)

The equation now says: squaring \((x-9)\) gives 25 as a result. This means \((x-9)\) must be 5 or -5.

\(\displaystyle \begin {align} x-9=5 \quad & \text{or} \quad x-9=\text-5\\ x=14 \quad & \text{or} \quad x=4 \end {align}\)