Lesson 14

Elena says, “\((x+3)^2\) can be expanded into \(x^2 + 6x + 9\). Likewise, \((2x+3)^2\) can be expanded into \(4x^2 + 6x + 9\).”

Find an error in Elena’s statement and correct the error. Show your reasoning.

1. Write each expression in standard form:

   1. \((4x+1)^2\)
   2. \((5x-2)^2\)
   3. \((\frac12 x + 7)^2\)
   4. \((3x+n)^2\)
   5. \((kx+m)^2\)

2. Decide if each expression is a perfect square. If so, write an equivalent expression of the form \((kx+m)^2\). If not, suggest one change to turn it into a perfect square.

   1. \(4x^2 + 12x + 9\)
   2. \(4x^2 + 8x+ 25\)

1. Find the value of \(c\) to make each expression in the left column a perfect square in standard form. Then, write an equivalent expression in the form of squared factors. In the last row, write your own pair of equivalent expressions.

   standard form \((ax^2+bx+c)\)	squared factors \((kx+m)^2\)
   \(100x^2+80x+c\)
   \(36x^2-60x+c\)
   \(25x^2+40x+c\)
   \(0.25x^2-14x+c\)

2. Solve each equation by completing the square:

   \(25x^2 + 40x = \text-12\)

   \(36x^2 - 60x + 10 = \text-6\)

Once you understand the methods, use each method at least one time to solve these equations.

1. \(5x^2 + 17x + 6 = 0\)
2. \(6x^2 + 19x = \text-10\)
3. \(8x^2 - 33x + 4 = 0\)
4. \(8x^2 - 26x = \text-21\)
5. \(10x^2 + 37x = 36\)
6. \(12x^2 + 20x - 77=0\)

In earlier lessons, we worked with perfect squares such as \((x+1)^2\) and \((x-5)(x-5)\). We learned that their equivalent expressions in standard form follow a predictable pattern:

• In general, \((x+m)^2\) can be written as \(x^2 + 2mx + m^2\).
• If a quadratic expression of the form \(ax^2 + bx + c\) is a perfect square, and the value of \(a\) is 1, then the value of \(b\) is \(2m\), and the value of \(c\) is \(m^2\) for some value of \(m\).

In this lesson, the variable in the factors being squared had a coefficient other than 1, for example \((3x+1)^2\) and \((2x-5)(2x-5)\). Their equivalent expression in standard form also followed the same pattern we saw earlier.

squared factors	standard form
\((3x+1)^2\)	\((3x)^2 + 2(3x)(1) + 1^2 \quad \text{or} \quad 9x^2 +6x + 1\)
\((2x-5)^2\)	\((2x)^2 + 2(2x)(\text-5) + (\text-5)^2 \quad \text{or} \quad 4x^2 -20x + 25\)

In general, \((kx+m)^2\) can be written as:

If a quadratic expression is of the form \(ax^2 + bx + c\), then:

• the value of \(a\) is \(k^2\)
• the value of \(b\) is \(2km\)
• the value of \(c\) is \(m^2\)

We can use this pattern to help us complete the square and solve equations when the squared term \(x^2\) has a coefficient other than 1—for example: \(16x^2 + 40x = 11\).

What constant term \(c\) can we add to make the expression on the left of the equal sign a perfect square? And how do we write this expression as squared factors?

• 16 is \(4^2\), so the squared factors could be \((4x + m)^2\).
• 40 is equal to \(2(4m)\), so \(2(4m) = 40\) or \(8m=40\). This means that \(m = 5\).
• If \(c\) is \(m^2\), then \(c = 5^2\) or \(c=25\).
• So the expression \(16x^2 + 40x + 25\) is a perfect square and is equivalent to \((4x+5)^2\).

Let’s solve the equation \(16x^2 + 40x = 11\) by completing the square!

\(\displaystyle \begin {align} 16x^2 + 40x &= 11\\ 16x^2 + 40x + 25 &= 11 + 25\\ (4x + 5)^2 &=36\\\\4x+5 = 6 \quad &\text {or} \quad 4x+5= \text-6\\ 4x = 1 \quad &\text {or} \quad 4x = \text-11\\ x=\frac14 \quad &\text {or} \quad x = \text- \frac{11}{4} \end {align}\).