Question

question 1 (multiple choice worth 1 points)
(08.03 mc)
solve for x using the quadratic formula: ( x^2 + 4x + 4 = 0 )
( x = \frac{-b pm sqrt{b^2 - 4ac}}{2a} )
( \bigcirc x = -2 )
( \bigcirc x = -1 )
( \bigcirc x = 0 )
( \bigcirc x = 1 )

question 2 (multiple choice worth 1 points)
(08.03 mc)
a portion of the quadratic formula proof is shown. fill in the missing reason.

statements	reasons
( ax^2 + bx = -c )	subtract c from both sides of the equation
( x^2 + \frac{b}{a}x = -\frac{c}{a} )	divide both sides of the equation by a

| ( x^2 + \frac{b}{a}x + left( \frac{b}{2a}
ight)^2 = -\frac{c}{a} + left( \frac{b}{2a}
ight)^2 ) | complete the square and add ( left( \frac{b}{2a}
ight)^2 ) to both sides |
| ( x^2 + \frac{b}{a}x + left( \frac{b}{2a}
ight)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} ) | square ( left( \frac{b}{2a}
ight) ) on the right side of the equation |
| ( x^2 + \frac{b}{a}x + left( \frac{b}{2a}
ight)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} ) |? |
( \bigcirc ) find a common denominator on the right side of the equation

Explanation:

Step1: Identify a, b, c values

From $x^2 + 4x + 4 = 0$, we get $a=1$, $b=4$, $c=4$.

Step2: Calculate discriminant

$\Delta = b^2 - 4ac = 4^2 - 4(1)(4) = 16 - 16 = 0$

Step3: Substitute into quadratic formula

$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-4 \pm \sqrt{0}}{2(1)} = \frac{-4}{2} = -2$

The right-hand side of the equation converts $-\frac{c}{a}$ to $-\frac{4ac}{4a^2}$ by multiplying numerator and denominator by $4a$, which creates a common denominator with $\frac{b^2}{4a^2}$. This action is finding a common denominator for the two terms on the right.

Answer:

x = -2

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