Factoring Any Quadratic Expression

To find an expression for the length $\ell ,$ the quadratic trinomial that represents the area will be factored. The first step to factoring the polynomial is to find its greatest common factor (GCF), if there is one. Note that the GCF of this expression is $3.$ The simplified quadratic trinomial in parentheses now has a leading coefficient of $1.$ Therefore, it can be factored accordingly. The expression $x^2+4x+3$ can be written as $(x+p)(x+q),$ where $p$ and $q$ are factor pairs of $3$ whose sum is $4.$

Factors of $3$	Sum of the Factors
$\text{-}1$ and $\text{-}3$	$\text{-}1+(\text{-}3)=\text{-}4\times$
$1$ and $3$	$1+3=4✓$

The table shows that $p=1$ and $q=3.$ With this information in mind, the expression can be fully factored. This expression represents the area of a tent's base. The expression $(x+3)$ on the right-hand side of the equation represents the width of the tent's base. A measurement for a width cannot be zero, which means that both sides of the equation can be divided by $(x+3).$ The length of the tent's base is $3x+3$ inches.

	Substitution	Calculation
$w=x+3$	$w=1+3$	$w=4$ feet
$\ell =3x+3$	$\ell =3\cdot 1+3$	$\ell =6$ feet

The class will buy tents that are $4$ feet wide and $6$ feet long.

To find an expression for the length $\ell ,$ the trinomial that represents the area will be factored. First, the values of $a,$ $b,$ and $c$ for the trinomial will be determined. Here, $a=3,$ $b=\text{-}5,$ and $c=2.$ With this in mind, the following conclusions can be made.

• The Greatest Common Factor (GCF) of $a,$ $b,$ and $c$ is $1.$
• Since $ac=6,$ which is greater than zero, the factors must have the same sign.
• Since $b=\text{-}5,$ which is negative, the sum of the factors will be negative. Therefore, both of the factors must be negative.

Now, all the negative factor pairs of $6$ will be listed and checked whether their sum is $\text{-}5.$

Factors of $ac=6$	Sum of Factors
$\text{-}1$ and $\text{-}6$	$\text{-}1+(\text{-}6)=\text{-}7$
$\text{-}2$ and $\text{-}3$	$\text{-}2+(\text{-}3)=\text{-}5$

Therefore, the $x\text{-}$term of the trinomial will be rewritten by using the factors $\text{-}2$ and $\text{-}3.$ Next, the expression will be factored by grouping its terms. Finally, the area can be written as the product of the binomials $x-1$ and $3x-2.$ Since $x-1$ represents the width of the pamphlet, the binomial $3x-2$ represents its length. Then, the length of the pamphlet will be found by substituting $x=5$ into the binomial that it was found in Part A. The length of the pamphlet is $13$ inches. The perimeter of the pamphlet can finally be calculated now that both the length and width are known. The perimeter of the pamphlet is $34$ inches.

An expression for the area of the tarp can be written by using the expressions for its length and width. Let $w$ be the width of the tarp. Since the length is $1$ inch less than twice the width, it can be expressed as $2w-1.$

Since the base is a rectangle, its area is equal to the product of its width and length. Now, the obtained quadratic expression will be factored. First, its coefficients will be identified. Here, $a=2,$ $b=\text{-}1,$ and $c=\text{-}28.$ With this in mind, the following conclusions can be made.

• The Greatest Common Factor (GCF) of $a,$ $b,$ and $c$ is $1.$
• Since $ac=\text{-}56,$ which is less than zero, the factors must have the different signs. Therefore, there is a positive factor and a negative factor.
• Since $b=\text{-}1,$ the sum of the factors is negative. This means that the absolute value of the negative factor is greater than the absolute value of the positive factor.

Now, the factor pairs of $\text{-}56$ with opposite signs and where the absolute value of the negative factor is greater than the absolute value of the positive number will be listed. Also, their sum will be calculated to see if it is $\text{-}1.$

Factors of $ac=\text{-}56$	Sum of Factors
$\text{-}56$ and $1$	$\text{-}56+1=\text{-}55$
$\text{-}28$ and $2$	$\text{-}28+2=\text{-}26$
$\text{-}14$ and $4$	$\text{-}14+4=\text{-}10$
$\text{-}8$ and $7$	$\text{-}8+7=\text{-}1$

Of these factors, $\text{-}8$ and $7$ satisfy the conditions. Therefore, the $x\text{-}$term of the trinomial on the left-hand side of the equation will be rewritten by using these numbers as a sum. Then, the obtained expression will be factored by grouping. Since the product of the two binomials is zero, at least one binomial must equal zero. With this in mind, the Zero Product Property will be applied. The solutions to the equation are $4$ and $\text{-}3.5.$ Since a negative length does not make sense, $w$ cannot be negative. This means that the value of the width is $4$ feet. With this information and knowing that the length is $1$ foot less than twice the width, the length can be found. The base of the tarp has a length and a width of $7$ and $4$ feet, respectively.

The height of the ball is zero when the it hits the ground. Therefore, to find the total time it takes for the ball to hit the ground, $0$ will be substituted for $h(t)$ in the given equation. Now, the trinomial will be factored. First, its coefficients will be identified. Here, $a=\text{-}5,$ $b=13,$ and $c=6.$ With this information in mind, the following conclusions can be made.

• The Greatest Common Factor (GCF) of $a,$ $b,$ and $c$ is $1.$
• Since $ac=\text{-}30,$ which is less than zero, the factors must have different signs. Therefore, one of the factors is positive and one of the factors is negative.
• Since $b=13,$ the sum of the factors is positive. Therefore, the absolute value of the positive factor is greater than the absolute value of the negative factor.

Now, the factor pairs of $\text{-}30$ with opposite signs and where the absolute value of the positive factor is greater than the absolute value of the negative factor will be listed. Also, their sum will be calculated to see whether it is $13.$

Factors of $ac=\text{-}30$	Sum of Factors
$\text{-}1$ and $30$	$29$
$\text{-}2$ and $15$	$13$
$\text{-}3$ and $10$	$7$
$\text{-}5$ and $6$	$1$

The factors of $\text{-}30$ whose sum is $13$ are $\text{-}2$ and $15.$ The $t\text{-}$term of the right-hand side can be rewritten as a subtraction of these numbers. Now, the obtained expression will be factored by grouping. The factoring process is finally done. Now, the Zero Product Property can be used to find the $t\text{-}$values. Since a negative time does not make sense, $t=3$ is the only solution that satisfies the conditions. This means that the ball will hit the ground $3$ seconds after it is thrown.

As seen above, the overall width is $x+30+x$ meters. The overall length is $x+50+x$ meters. Since the given figure is a rectangle, its area is the product of its width and length. The product will be rewritten as a trinomial. The trinomial that represents the total area is $4x^2+160x+1500.$ The equation will be rewritten to make the right-hand side equal to zero. Next, the Greatest Common Factor (GCF) will be factored out. Now that the GCF has been factored out, the quadratic trinomial in parentheses will be factored. The expression $x^2+40x-176$ will be written as $(x+p)(x+q),$ where $p$ and $q$ are factor pairs of $\text{-}176$ whose sum is $40.$ Therefore, all factor pairs of $\text{-}176$ will be listed and their sum will be calculated.

Factors of $\text{-}176$	Sum of Factors
$\text{-}1$ and $176$	$\text{-}1+176=175$
$\text{-}2$ and $88$	$\text{-}2+88=86$
$\text{-}4$ and $44$	$\text{-}4+44=40$
$\text{-}8$ and $22$	$\text{-}8+22=14$
$\text{-}11$ and $16$	$\text{-}11+16=5$
$\text{-}16$ and $11$	$\text{-}16+11=\text{-}5$
$\text{-}22$ and $8$	$\text{-}22+8=\text{-}14$
$\text{-}44$ and $4$	$\text{-}44+4=\text{-}40$
$\text{-}88$ and $2$	$\text{-}88+2=\text{-}86$
$\text{-}176$ and $1$	$\text{-}176+1=\text{-}175$

As seen, the factors $\text{-}4$ and $44$ satisfy the conditions, so $p=\text{-}4$ and $q=44.$ With this information in mind, the equation can now be fully factored. Finally, since the product of the two binomials is zero, at least one binomial must equal zero. With this in mind, the Zero Product Property can be used to find the value of $x.$ Since $x$ represents the width of the track, it cannot be negative. Therefore, the only solution to the equation is $x=4.$ This means that the width of the jogging track is $4$ meters.