12 multiple-choice questions, progressively harder.
Solve 2x2+4x−3=02x^2 + 4x - 3 = 02x2+4x−3=0.
Solution
Correct answer: C
Factor 222 and complete the square: 2x2+4x−3=2(x+1)2−52x^2 + 4x - 3 = 2(x + 1)^2 - 52x2+4x−3=2(x+1)2−5.
2(x+1)2−5=0 ⇒ (x+1)2=522(x + 1)^2 - 5 = 0 \ \Rightarrow\ (x + 1)^2 = \tfrac{5}{2}2(x+1)2−5=0 ⇒ (x+1)2=25
Taking the root, x+1=±52=±102x + 1 = \pm\sqrt{\tfrac{5}{2}} = \pm\dfrac{\sqrt{10}}{2}x+1=±25=±210, so x=−1±102x = -1 \pm \dfrac{\sqrt{10}}{2}x=−1±210.
What is the vertex of f(x)=x2−12x+40f(x) = x^2 - 12x + 40f(x)=x2−12x+40?
Correct answer: B
Complete the square: half of −12-12−12 is −6-6−6, and (−6)2=36(-6)^2 = 36(−6)2=36.
x2−12x+40=(x−6)2−36+40=(x−6)2+4x^2 - 12x + 40 = (x - 6)^2 - 36 + 40 = (x - 6)^2 + 4x2−12x+40=(x−6)2−36+40=(x−6)2+4
Vertex form gives vertex (6,4)(6, 4)(6,4).
What is the range of f(x)=−2x2+8x−1f(x) = -2x^2 + 8x - 1f(x)=−2x2+8x−1?
Correct answer: D
Factor −2-2−2 from the first two terms, then complete the square.
−2x2+8x−1=−2(x2−4x)−1=−2(x−2)2+8−1=−2(x−2)2+7-2x^2 + 8x - 1 = -2(x^2 - 4x) - 1 = -2(x - 2)^2 + 8 - 1 = -2(x - 2)^2 + 7−2x2+8x−1=−2(x2−4x)−1=−2(x−2)2+8−1=−2(x−2)2+7
The leading coefficient is negative, so the maximum value is 777 and the range is y≤7y \le 7y≤7.
Solve x2−2x−4=0x^2 - 2x - 4 = 0x2−2x−4=0.
Half of −2-2−2 is −1-1−1, and (−1)2=1(-1)^2 = 1(−1)2=1.
x2−2x−4=(x−1)2−1−4=(x−1)2−5x^2 - 2x - 4 = (x - 1)^2 - 1 - 4 = (x - 1)^2 - 5x2−2x−4=(x−1)2−1−4=(x−1)2−5
Set to zero: (x−1)2=5(x - 1)^2 = 5(x−1)2=5, so x−1=±5x - 1 = \pm\sqrt{5}x−1=±5 and x=1±5x = 1 \pm \sqrt{5}x=1±5.
The minimum of x2+bx+12x^2 + bx + 12x2+bx+12 occurs at x=3x = 3x=3. What is bbb?
Completing the square puts the vertex at x=−b2x = -\tfrac{b}{2}x=−2b, and the minimum occurs there.
−b2=3 ⇒ b=−6-\tfrac{b}{2} = 3 \ \Rightarrow\ b = -6−2b=3 ⇒ b=−6
With b=−6b = -6b=−6 the expression is x2−6x+12=(x−3)2+3x^2 - 6x + 12 = (x - 3)^2 + 3x2−6x+12=(x−3)2+3, whose minimum is at x=3x = 3x=3.
Solve (2x−1)2=9(2x - 1)^2 = 9(2x−1)2=9.
Take the square root of both sides with a ±\pm±.
2x−1=±3 ⇒ 2x=1±32x - 1 = \pm 3 \ \Rightarrow\ 2x = 1 \pm 32x−1=±3 ⇒ 2x=1±3
So 2x=42x = 42x=4 or 2x=−22x = -22x=−2, giving x=2x = 2x=2 or x=−1x = -1x=−1.
A quadratic f(x)=x2+bx+cf(x) = x^2 + bx + cf(x)=x2+bx+c has vertex (2,−3)(2, -3)(2,−3). What is ccc?
Correct answer: A
Vertex (2,−3)(2, -3)(2,−3) means f(x)=(x−2)2−3f(x) = (x - 2)^2 - 3f(x)=(x−2)2−3. Expand to read the constant term.
(x−2)2−3=x2−4x+4−3=x2−4x+1(x - 2)^2 - 3 = x^2 - 4x + 4 - 3 = x^2 - 4x + 1(x−2)2−3=x2−4x+4−3=x2−4x+1
Matching x2+bx+cx^2 + bx + cx2+bx+c gives c=1c = 1c=1 (and b=−4b = -4b=−4).
How many real solutions does x2−6x+9=0x^2 - 6x + 9 = 0x2−6x+9=0 have?
The left side is a perfect square, since half of −6-6−6 squared is 999.
x2−6x+9=(x−3)2=0 ⇒ x=3x^2 - 6x + 9 = (x - 3)^2 = 0 \ \Rightarrow\ x = 3x2−6x+9=(x−3)2=0 ⇒ x=3
The roots coincide, so there is exactly one real solution.
The expression x2−10x+30x^2 - 10x + 30x2−10x+30 has minimum value mmm reached at x=px = px=p. What is m+pm + pm+p?
Complete the square to reach vertex form.
x2−10x+30=(x−5)2−25+30=(x−5)2+5x^2 - 10x + 30 = (x - 5)^2 - 25 + 30 = (x - 5)^2 + 5x2−10x+30=(x−5)2−25+30=(x−5)2+5
The minimum value is m=5m = 5m=5, reached at p=5p = 5p=5, so m+p=10m + p = 10m+p=10.
For x2+kx+25x^2 + kx + 25x2+kx+25 to be a perfect square trinomial, what is kkk?
A perfect square needs (k2)2=25\left(\tfrac{k}{2}\right)^2 = 25(2k)2=25.
k2=±5 ⇒ k=±10\tfrac{k}{2} = \pm 5 \ \Rightarrow\ k = \pm 102k=±5 ⇒ k=±10
Both work: x2+10x+25=(x+5)2x^2 + 10x + 25 = (x + 5)^2x2+10x+25=(x+5)2 and x2−10x+25=(x−5)2x^2 - 10x + 25 = (x - 5)^2x2−10x+25=(x−5)2.
Written in vertex form, 12x2+2x+1\tfrac{1}{2}x^2 + 2x + 121x2+2x+1 equals which of these?
Factor 12\tfrac{1}{2}21 from the first two terms, then complete the square inside.
12x2+2x+1=12(x2+4x)+1=12((x+2)2−4)+1\tfrac{1}{2}x^2 + 2x + 1 = \tfrac{1}{2}(x^2 + 4x) + 1 = \tfrac{1}{2}\big((x + 2)^2 - 4\big) + 121x2+2x+1=21(x2+4x)+1=21((x+2)2−4)+1
Distribute: 12(x+2)2−2+1=12(x+2)2−1\tfrac{1}{2}(x + 2)^2 - 2 + 1 = \tfrac{1}{2}(x + 2)^2 - 121(x+2)2−2+1=21(x+2)2−1.
What is the vertex of f(x)=2x2−8x+1f(x) = 2x^2 - 8x + 1f(x)=2x2−8x+1?
Factor 222 from the first two terms, then complete the square.
2x2−8x+1=2(x2−4x)+1=2(x−2)2−8+1=2(x−2)2−72x^2 - 8x + 1 = 2(x^2 - 4x) + 1 = 2(x - 2)^2 - 8 + 1 = 2(x - 2)^2 - 72x2−8x+1=2(x2−4x)+1=2(x−2)2−8+1=2(x−2)2−7
Vertex form gives vertex (2,−7)(2, -7)(2,−7).
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