Lesson 11

What are Perfect Squares?

  • Let’s see how perfect squares make some equations easier to solve.

11.1: The Thing We Are Squaring

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. \((2x-9)(2x-9)=a^2\)

11.2: Perfect Squares in Different Forms

  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?

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



Write each expression in factored form.

  1. \(x^4-30x^2+225\)
  2. \(x+14\sqrt{x}+49\)
  3. \(5^{2x}+6 \boldcdot 5^x + 9\)

11.3: Two Methods

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

Han’s method:

\(\displaystyle \begin {align} (x-6)^2&=25\\(x-6)(x-6)&=25 \\x^2-12x+36&=25\\ x^2-12x+11&=0\\(x-11)(x-1)&=0\\ \\x=11 \quad \text{or} \quad x&=1 \end{align}\)

Jada’s method:

\(\displaystyle \begin {align} (x-6)^2&=25\\ \\x-6=5 \quad &\text{or} \quad x-6=\text-5\\ x=11 \quad &\text{or} \quad x=1 \end{align}\)

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.

\((y-5)^2=49\)

\((x+4)^2=9\)

\((z+\frac13)^2=\frac49\)

\((v - 0.1)^2=0.36\)

 

Summary

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.

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

Glossary Entries

  • perfect square

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

  • rational number

    A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into \(b\) equal parts and finding the point that is \(a\) of them from 0. We can always write a fraction in the form \(\frac{a}{b}\) where \(a\) and \(b\) are whole numbers, with \(b\) not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as \(\frac{7}{10}\).

    The numbers \(3\), \(\text-\frac34\), and \(6.7\) are all rational numbers. The numbers \(\pi\) and \(\text-\sqrt{2}\) are not rational numbers, because they cannot be written as fractions or their opposites.