12 multiple-choice questions, progressively harder.
Solve x2−6x+11=0x^2 - 6x + 11 = 0x2−6x+11=0.
Solution
Correct answer: A
Read off a=1a = 1a=1, b=−6b = -6b=−6, c=11c = 11c=11, so −b=6-b = 6−b=6 and D=36−44=−8D = 36 - 44 = -8D=36−44=−8, with −8=2i2\sqrt{-8} = 2i\sqrt{2}−8=2i2.
x=6±2i22=3±i2x = \frac{6 \pm 2i\sqrt{2}}{2} = 3 \pm i\sqrt{2}x=26±2i2=3±i2
Divide both parts of the numerator by 222.
Solve x2+4x+13=0x^2 + 4x + 13 = 0x2+4x+13=0.
Correct answer: D
Read off a=1a = 1a=1, b=4b = 4b=4, c=13c = 13c=13, so D=16−52=−36D = 16 - 52 = -36D=16−52=−36, with −36=6i\sqrt{-36} = 6i−36=6i.
x=−4±6i2=−2±3ix = \frac{-4 \pm 6i}{2} = -2 \pm 3ix=2−4±6i=−2±3i
Divide both the −4-4−4 and the 6i6i6i by 222.
Solve 4x2−8x+3=04x^2 - 8x + 3 = 04x2−8x+3=0.
Correct answer: C
Read off a=4a = 4a=4, b=−8b = -8b=−8, c=3c = 3c=3, so −b=8-b = 8−b=8 and D=64−48=16D = 64 - 48 = 16D=64−48=16.
x=8±168=8±48x = \frac{8 \pm \sqrt{16}}{8} = \frac{8 \pm 4}{8}x=88±16=88±4
So x=128=32x = \frac{12}{8} = \tfrac{3}{2}x=812=23 or x=48=12x = \frac{4}{8} = \tfrac{1}{2}x=84=21.
Solve x2−2x−1=0x^2 - 2x - 1 = 0x2−2x−1=0.
Read off a=1a = 1a=1, b=−2b = -2b=−2, c=−1c = -1c=−1, so −b=2-b = 2−b=2 and −4ac=+4-4ac = +4−4ac=+4, giving D=4+4=8D = 4 + 4 = 8D=4+4=8, with 8=22\sqrt{8} = 2\sqrt{2}8=22.
x=2±222=1±2x = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2}x=22±22=1±2
For what value of ccc does 2x2+4x+c=02x^2 + 4x + c = 02x2+4x+c=0 have two complex roots?
Correct answer: B
Complex roots need D<0D < 0D<0, where D=42−4(2)c=16−8cD = 4^2 - 4(2)c = 16 - 8cD=42−4(2)c=16−8c.
16−8c<0⇒c>216 - 8c < 0 \quad\Rightarrow\quad c > 216−8c<0⇒c>2
Of the choices only c=3c = 3c=3 exceeds 222. At c=2c = 2c=2 the discriminant is 000 (a repeated real root), and smaller ccc gives two real roots.
Solve x2−4x+7=0x^2 - 4x + 7 = 0x2−4x+7=0.
Read off a=1a = 1a=1, b=−4b = -4b=−4, c=7c = 7c=7, so −b=4-b = 4−b=4 and D=16−28=−12D = 16 - 28 = -12D=16−28=−12, with −12=2i3\sqrt{-12} = 2i\sqrt{3}−12=2i3.
x=4±2i32=2±i3x = \frac{4 \pm 2i\sqrt{3}}{2} = 2 \pm i\sqrt{3}x=24±2i3=2±i3
Solve x2+6x+13=0x^2 + 6x + 13 = 0x2+6x+13=0.
Read off a=1a = 1a=1, b=6b = 6b=6, c=13c = 13c=13, so D=36−52=−16D = 36 - 52 = -16D=36−52=−16, with −16=4i\sqrt{-16} = 4i−16=4i.
x=−6±4i2=−3±2ix = \frac{-6 \pm 4i}{2} = -3 \pm 2ix=2−6±4i=−3±2i
Solve 2x2−6x+1=02x^2 - 6x + 1 = 02x2−6x+1=0.
Read off a=2a = 2a=2, b=−6b = -6b=−6, c=1c = 1c=1, so −b=6-b = 6−b=6 and D=36−8=28D = 36 - 8 = 28D=36−8=28, with 28=27\sqrt{28} = 2\sqrt{7}28=27.
x=6±274=2(3±7)4=3±72x = \frac{6 \pm 2\sqrt{7}}{4} = \frac{2(3 \pm \sqrt{7})}{4} = \frac{3 \pm \sqrt{7}}{2}x=46±27=42(3±7)=23±7
Reduce numerator and denominator by the common factor 222.
How many real roots does 9x2+12x+4=09x^2 + 12x + 4 = 09x2+12x+4=0 have?
Compute the discriminant with a=9a = 9a=9, b=12b = 12b=12, c=4c = 4c=4.
D=122−4(9)(4)=144−144=0D = 12^2 - 4(9)(4) = 144 - 144 = 0D=122−4(9)(4)=144−144=0
A zero discriminant means one repeated real root, x=−23x = -\tfrac{2}{3}x=−32, since 9x2+12x+4=(3x+2)29x^2 + 12x + 4 = (3x + 2)^29x2+12x+4=(3x+2)2.
Solve x2+2x−6=0x^2 + 2x - 6 = 0x2+2x−6=0.
Read off a=1a = 1a=1, b=2b = 2b=2, c=−6c = -6c=−6, so −4ac=+24-4ac = +24−4ac=+24 and D=4+24=28D = 4 + 24 = 28D=4+24=28, with 28=27\sqrt{28} = 2\sqrt{7}28=27.
x=−2±272=−1±7x = \frac{-2 \pm 2\sqrt{7}}{2} = -1 \pm \sqrt{7}x=2−2±27=−1±7
Solve x2−5x+5=0x^2 - 5x + 5 = 0x2−5x+5=0.
Read off a=1a = 1a=1, b=−5b = -5b=−5, c=5c = 5c=5, so −b=5-b = 5−b=5 and D=25−20=5D = 25 - 20 = 5D=25−20=5.
x=5±52x = \frac{5 \pm \sqrt{5}}{2}x=25±5
The discriminant 555 has no square factor to pull out, so the radical stays as 5\sqrt{5}5 and nothing reduces.
Solve 2x2+3x−2=02x^2 + 3x - 2 = 02x2+3x−2=0.
Read off a=2a = 2a=2, b=3b = 3b=3, c=−2c = -2c=−2, so −4ac=+16-4ac = +16−4ac=+16 and D=9+16=25D = 9 + 16 = 25D=9+16=25.
x=−3±254=−3±54x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}x=4−3±25=4−3±5
So x=24=12x = \frac{2}{4} = \tfrac{1}{2}x=42=21 or x=−84=−2x = \frac{-8}{4} = -2x=4−8=−2.
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