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Simplify the following expressions: a) (z-7) b1z-a c) z-12

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5 parts
a

z2−5z+10z^2 - 5z + 10

b

Sum=5, Product=6\text{Sum} = 5, \text{ Product} = 6

c

x=12,−2x = \dfrac{1}{2}, -2

d

x=−2x = -2

e

x=13,−1x = \dfrac{1}{3}, -1

Step-by-step solution

14 steps

Part a· 3 steps

(z−1)2−3(z−3)(z-1)^2 - 3(z-3)

  1. Expand (z−1)2(z-1)^2.

    (z−1)2=z2−2z+1(z-1)^2 = z^2 - 2z + 1

  2. Expand 3(z−3)3(z-3).

    3(z−3)=3z−93(z-3) = 3z - 9

  3. Subtract.

    z2−2z+1−(3z−9)=z2−2z+1−3z+9z^2 - 2z + 1 - (3z - 9) = z^2 - 2z + 1 - 3z + 9 =z2−5z+10= z^2 - 5z + 10

    Simplified: z2−5z+10z^2 - 5z + 10

    z2−5z+10\boxed{z^2 - 5z + 10}

Part b

Sum=5, Product=6\text{Sum} = 5, \text{ Product} = 6

x2−5x+6=0x^2 - 5x + 6 = 0

  1. Sum of roots formula

    −ba-\frac{b}{a}.
    a=1a=1, b=−5b=-5

    −−51=5-\frac{-5}{1} = 5

  2. Product of roots formula

    ca\frac{c}{a}.
    c=6c=6

    61=6\frac{6}{1} = 6

    Sum =5= 5, Product =6= 6

    Sum=5, Product=6\boxed{\text{Sum} = 5, \text{ Product} = 6}

Part c

x=12,−2x = \dfrac{1}{2}, -2

2x2+3x−2=02x^2 + 3x - 2 = 0

  1. Quadratic formula

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
    a=2a=2, b=3b=3, c=−2c=-2

  2. Discriminant D=b2−4acD = b^2 - 4ac.

    D=32−4(2)(−2)=9+16=25D = 3^2 - 4(2)(-2) = 9 + 16 = 25 D=5\sqrt{D} = 5

  3. Roots.

    x=−3±54x = \frac{-3 \pm 5}{4} x1=−3+54=24=12x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2} x2=−3−54=−84=−2x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2

    Roots: x=12,−2x = \frac{1}{2}, -2

    x=12,−2\boxed{x = \dfrac{1}{2}, -2}

Part d

x=−2x = -2

x2+4x+4=0x^2 + 4x + 4 = 0

  1. Quadratic formula

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
    a=1a=1, b=4b=4, c=4c=4

  2. Discriminant D=b2−4acD = b^2 - 4ac.

    D=16−16=0D = 16 - 16 = 0 D=0\sqrt{D} = 0

  3. Root.

    x=−4±02=−42=−2x = \frac{-4 \pm 0}{2} = \frac{-4}{2} = -2

    Root: x=−2x = -2 (repeated)

    x=−2\boxed{x = -2}

Part e

x=13,−1x = \dfrac{1}{3}, -1

3x2+2x−1=03x^2 + 2x - 1 = 0

  1. Quadratic formula

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
    a=3a=3, b=2b=2, c=−1c=-1

  2. Discriminant D=b2−4acD = b^2 - 4ac.

    D=4−4(3)(−1)=4+12=16D = 4 - 4(3)(-1) = 4 + 12 = 16 D=4\sqrt{D} = 4

  3. Roots.

    x=−2±46x = \frac{-2 \pm 4}{6} x1=−2+46=26=13x_1 = \frac{-2 + 4}{6} = \frac{2}{6} = \frac{1}{3} x2=−2−46=−66=−1x_2 = \frac{-2 - 4}{6} = \frac{-6}{6} = -1

    Roots: x=13,−1x = \frac{1}{3}, -1

    x=13,−1\boxed{x = \dfrac{1}{3}, -1}
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