9.4.3 Standard Form and Squared Factors

Activity

1. For each expression, find the value of $c$ to make a perfect square in standard form.

2. For each expression, refer to your responses in question 1 to help write an equivalent expression in the form of squared factors.

3. Write your own pair of equivalent expressions.

4. Solve each equation by completing the square.

Find the Value to Make a Perfect Square and Solve

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 $a{x}^2+bx+c$ 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.

In general, $(kx+m{)}^2$ can be written as:

$(kx{)}^2+2(kx)(m)+m^2$ or $k^2{x}^2+2kmx+m^2$

If a quadratic expression is of the form $a{x}^2+bx+c$, 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!

Step 1 -

$16x^2+40x=11$

To be a perfect square, $(4x+m{)}^2$ is needed.

We found the $4x$ by determining the square root of the first term, $16x^2$.

Step 2 -

Now, find the value of $m$ to make a perfect square.

Since $b=2km$, and we know that $k=4$ (from step 1), and that $b=40$ (from the coefficient of the linear term), we substitute the values and solve.

$40=2(4m)$

$40=8m,$

so $m=5.$

Step 3 -

Find $c=m^2.$

$c=m^2=5^2=25$

Step 4 -

Add $c$ to both sides of the equation.

$16x^2+40x+25=11+25$

$16x^2+40x+25=36$

Step 5 -

Factor and simplify.

$(4x+5{)}^2=6^2$

Step 6 -

Take the square root of both sides. Remember the plus or minus.

$(4x+5{)}^2=6^2$

$\sqrt{(4x+5{)}^2}=\sqrt{6^2}$

$4x+5=\pm 6$

Step 7 -

Write two equations and solve.

$4x+5=6$      or    $4x+5=-6$

          $4x=1$    or    $4x=-11$

          $x=\frac{1}{4}$      or    $x=-\frac{11}{4}$