CPM Homework Help : INT2 Problem 5-118

Solve the following quadratic equations using any method.

$10000x^2−64=0$

$9x^2−8=−34x$

$2x^2−4x+7=0$

$3.2x+0.2x^2−5=0$

Hint:

Example 1: Solve $3x^2+x−14=0$ for $x$ using the Zero Product Property.

Solution: First, factor the quadratic so it is written as a product: $(3x+7)(x−2)=0$ . (If factoring is not possible, one of the other methods of solving must be used.) The Zero Product Property states that if the product of two terms is $0$ , then at least one of the factors must be $0$ . Thus, $3x+7=0$ or $x−2=0$ . Solving these equations for $x$ reveals that $x=\frac{7}{3}$ or that $x=2$ .

Example 2: Solve $3x^2+x−14=0$ for $x$ using the Quadratic Formula.

Solution: This method works for any quadratic. First, identify $a,b$ , and $c$ . $a$ equals the number of $x^2$ -terms, $b$ equals the number of $x$ -terms, and $c$ equals the constant. For $3x^2+x−14=0,a=3,b=1$ , and $c=−14$ . Substitute the values of $a,b$ , and $c$ into the Quadratic Formula and evaluate the expression twice: once with addition and once with subtraction. Examine this method below:

$x=\frac{-1+\sqrt{1^2-4(3)(-14)}} {2⋅3}$		$x=\frac{-1+\sqrt{1^2-4(3)(-14)}} {2⋅3}$
$=\frac{-1+\sqrt{169}} {6}$	or	$=\frac{-1-\sqrt{169}}{6}$
$=\frac{12}{6}=2$		$=\frac{-14}{6}=-\frac{7}{3}$

Example 3: Solve $x^2+5x+4=0$ by completing the square.

Solution: This method works most efficiently when the coefficient of $x^2$ is $1$ . Rewrite the equation as $x^2+5x=−4$ . Rewrite the left side as an incomplete square:

Generic rectangle left edge $2.5$

$x$

interior top left $2.5x$

blank

interior bottom left $x^2$

interior bottom right $2.5x$

right edge $=-4$

bottom edge $x\ +\ 2.5$

Complete the square and rewrite as
$(x+2.5)^2−6.25=−4$ or

$(x+2.5)^2=2.25$

Take the square root of both sides, $x+2.5=\pm1.5$ . Solving for $x$ reveals that $x=−1$ or $x=−4$ .