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To find the discriminant and describe the nature of the roots for each quadratic equation, we use the formula for the discriminant = b² − 4ac, where the quadratic equation is in the form $ax

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5 parts
a
4
b
0
c
-16
d
0
e
9

Step-by-step solution

15 steps

To find the discriminant and describe the nature of the roots for each quadratic equation, we use the formula for the discriminant Δ=b2−4ac\Delta = b^2 - 4ac, where the quadratic equation is in the form ax2+bx+c=0ax^2 + bx + c = 0.

The nature of the roots is determined as follows:

  • If Δ>0\Delta > 0, there are two distinct real roots.
  • If Δ=0\Delta = 0, there is one real root (a repeated root).
  • If Δ<0\Delta < 0, there are two complex conjugate roots (no real roots).

Part a· 3 steps

x2−4x+3=0x^2 - 4x + 3 = 0

  1. Identify $a, b, c$.

    Here, a=1a = 1, b=−4b = -4, c=3c = 3.

  2. Calculate the discriminant.

    Δ=b2−4ac=(−4)2−4(1)(3)\Delta = b^2 - 4ac = (-4)^2 - 4(1)(3) Δ=16−12\Delta = 16 - 12 Δ=4\Delta = 4

  3. Describe the nature of the roots.

    Since Δ=4>0\Delta = 4 > 0, the equation has two distinct real roots.
    The discriminant is 4\boxed{4}.

Part b0

x2−6x+9=0x^2 - 6x + 9 = 0

  1. Identify $a, b, c$.

    Here, a=1a = 1, b=−6b = -6, c=9c = 9.

  2. Calculate the discriminant.

    Δ=b2−4ac=(−6)2−4(1)(9)\Delta = b^2 - 4ac = (-6)^2 - 4(1)(9) Δ=36−36\Delta = 36 - 36 Δ=0\Delta = 0

  3. Describe the nature of the roots.

    Since Δ=0\Delta = 0, the equation has one real root (a repeated root).
    The discriminant is 0\boxed{0}.

Part c-16

x2+2x+5=0x^2 + 2x + 5 = 0

  1. Identify $a, b, c$.

    Here, a=1a = 1, b=2b = 2, c=5c = 5.

  2. Calculate the discriminant.

    Δ=b2−4ac=(2)2−4(1)(5)\Delta = b^2 - 4ac = (2)^2 - 4(1)(5) Δ=4−20\Delta = 4 - 20 Δ=−16\Delta = -16

  3. Describe the nature of the roots.

    Since Δ=−16<0\Delta = -16 < 0, the equation has two complex conjugate roots (no real roots).
    The discriminant is −16\boxed{-16}.

Part d0

3x2−12x+12=03x^2 - 12x + 12 = 0

  1. Identify $a, b, c$.

    Here, a=3a = 3, b=−12b = -12, c=12c = 12.

  2. Calculate the discriminant.

    Δ=b2−4ac=(−12)2−4(3)(12)\Delta = b^2 - 4ac = (-12)^2 - 4(3)(12) Δ=144−144\Delta = 144 - 144 Δ=0\Delta = 0

  3. Describe the nature of the roots.

    Since Δ=0\Delta = 0, the equation has one real root (a repeated root).
    The discriminant is 0\boxed{0}.

Part e9

x2−7x+10=0x^2 - 7x + 10 = 0

  1. Identify $a, b, c$.

    Here, a=1a = 1, b=−7b = -7, c=10c = 10.

  2. Calculate the discriminant.

    Δ=b2−4ac=(−7)2−4(1)(10)\Delta = b^2 - 4ac = (-7)^2 - 4(1)(10) Δ=49−40\Delta = 49 - 40 Δ=9\Delta = 9

  3. Describe the nature of the roots.

    Since Δ=9>0\Delta = 9 > 0, the equation has two distinct real roots.
    The discriminant is 9\boxed{9}.

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