Question

algebra 2: re do quiz #3
unit 2.3 : solving and applying quadratic equations
name: date: period: score:
directions: read each statement carefully. choose and circle the letter of the correct
answer. show your work if necessary!
section 1: no calculator

1. selected response: 1 point for correct answer

$f(x)=2x^{2}-2x - 24$
which of the following is not a factor of the given quadratic function?
a. 2
b. $x - 6$
c. $x - 4$
d. $x + 3$

1. selected response: 1 point for correct answer

$(x + 5)^{2}+11 = 2$
how many distinct real roots do the given equation have?
a. none
b. exactly one
c. exactly two
d. infinitely many
selected response: 1 point for correct answer
$x^{2}+8x - 3$
which of the following is equivalent to the expression above?
a. $(x + 4)^{2}-19$
b. $(x + 4)^{2}+13$
c. $(x - 4)^{2}-19$
d. $(x - 4)^{2}+13$

Explanation:

Step1: Factor the quadratic function

Factor \( f(x)=2x^{2}-2x - 24\). First, factor out the common factor \(2\): \(f(x)=2(x^{2}-x - 12)\). Then factor the quadratic \(x^{2}-x - 12=(x - 4)(x+3)\). So \(f(x)=2(x - 4)(x + 3)\).

Step2: Analyze each option

• Option A: \(2\) is a factor.
• Option B: If \(x-6\) is a factor, then \(f(6)\) should be \(0\). \(f(6)=2\times6^{2}-2\times6-24=2\times36-12 - 24=72-12 - 24 = 36

eq0\).

• Option C: \(x - 4\) is a factor as shown in Step1.
• Option D: \(x + 3\) is a factor as shown in Step1.

Step1: Solve the equation \((x + 5)^{2}+11=2\)

Subtract \(11\) from both sides: \((x + 5)^{2}=2-11=-9\).

Step2: Analyze the nature of the roots

Since the square of a real number \((x + 5)^{2}\geq0\) for all real \(x\), and \(-9<0\), there are no real values of \(x\) that satisfy the equation \((x + 5)^{2}=-9\).

Step1: Complete the square for \(x^{2}+8x-3\)

For the quadratic \(ax^{2}+bx + c\) (here \(a = 1\), \(b = 8\), \(c=-3\)), the formula for completing the square is \((x+\frac{b}{2})^{2}-\frac{b^{2}}{4}+c\).
\(x^{2}+8x-3=(x + 4)^{2}-16-3\) (because \(\frac{b}{2}=\frac{8}{2}=4\) and \((\frac{b}{2})^{2}=16\)).

Step2: Simplify the expression

\((x + 4)^{2}-16-3=(x + 4)^{2}-19\)

Answer:

B. \(x - 6\)