Math Problem Statement

Roger writes five quadratic equations and tells you that two of them have complex roots. However, he has written the equations in vertex form, expecting that you will have to rewrite the equations in standard form so you can calculate the discriminants.

Which TWO of Roger’s equations have complex roots?

A.y=(x−3)2+3y=(x−3)2+3

B.y=−2(x+4)2+5y=−2(x+4)2+5

C.y=13(x−7)2−10y=13(x−7)2−10

D.y=−5(x−12)2y=−5(x−12)2

E.y=−12(x+1)2−2y=−12(x+1)2−2

Solution

To determine which of Roger's quadratic equations have complex roots, we need to rewrite each equation in standard form and calculate its discriminant (Δ=b24ac\Delta = b^2 - 4ac).

The vertex form of a quadratic equation is:

y=a(xh)2+ky = a(x - h)^2 + k

Rewriting into standard form y=ax2+bx+cy = ax^2 + bx + c, and then calculating the discriminant for each:


A. y=(x3)2+3y = (x - 3)^2 + 3

  1. Expand: y=(x3)2+3=x26x+9+3=x26x+12y = (x - 3)^2 + 3 = x^2 - 6x + 9 + 3 = x^2 - 6x + 12 a=1,b=6,c=12a = 1, b = -6, c = 12

  2. Discriminant: Δ=b24ac=(6)24(1)(12)=3648=12\Delta = b^2 - 4ac = (-6)^2 - 4(1)(12) = 36 - 48 = -12 Δ<0\Delta < 0, so roots are complex.


B. y=2(x+4)2+5y = -2(x + 4)^2 + 5

  1. Expand: y=2(x+4)2+5=2(x2+8x+16)+5=2x216x32+5=2x216x27y = -2(x + 4)^2 + 5 = -2(x^2 + 8x + 16) + 5 = -2x^2 - 16x - 32 + 5 = -2x^2 - 16x - 27 a=2,b=16,c=27a = -2, b = -16, c = -27

  2. Discriminant: Δ=b24ac=(16)24(2)(27)=256216=40\Delta = b^2 - 4ac = (-16)^2 - 4(-2)(-27) = 256 - 216 = 40 Δ>0\Delta > 0, so roots are real.


C. y=13(x7)210y = \frac{1}{3}(x - 7)^2 - 10

  1. Expand: y=13(x7)210=13(x214x+49)10=13x2143x+49310y = \frac{1}{3}(x - 7)^2 - 10 = \frac{1}{3}(x^2 - 14x + 49) - 10 = \frac{1}{3}x^2 - \frac{14}{3}x + \frac{49}{3} - 10 Combine constants: y=13x2143x493y = \frac{1}{3}x^2 - \frac{14}{3}x - \frac{49}{3} a=13,b=143,c=493a = \frac{1}{3}, b = -\frac{14}{3}, c = -\frac{49}{3}

  2. Discriminant: Δ=b24ac=(143)24(13)(493)\Delta = b^2 - 4ac = \left(-\frac{14}{3}\right)^2 - 4\left(\frac{1}{3}\right)\left(-\frac{49}{3}\right) Δ=19691969=1969+1969=3929\Delta = \frac{196}{9} - \frac{-196}{9} = \frac{196}{9} + \frac{196}{9} = \frac{392}{9} Δ>0\Delta > 0, so roots are real.


D. y=5(x12)2y = -5(x - 12)^2

  1. Expand: y=5(x12)2=5(x224x+144)=5x2+120x720y = -5(x - 12)^2 = -5(x^2 - 24x + 144) = -5x^2 + 120x - 720 a=5,b=120,c=720a = -5, b = 120, c = -720

  2. Discriminant: Δ=b24ac=(120)24(5)(720)=1440014400=0\Delta = b^2 - 4ac = (120)^2 - 4(-5)(-720) = 14400 - 14400 = 0 Δ=0\Delta = 0, so roots are real and repeated.


E. y=12(x+1)22y = -12(x + 1)^2 - 2

  1. Expand: y=12(x+1)22=12(x2+2x+1)2=12x224x122=12x224x14y = -12(x + 1)^2 - 2 = -12(x^2 + 2x + 1) - 2 = -12x^2 - 24x - 12 - 2 = -12x^2 - 24x - 14 a=12,b=24,c=14a = -12, b = -24, c = -14

  2. Discriminant: Δ=b24ac=(24)24(12)(14)=576672=96\Delta = b^2 - 4ac = (-24)^2 - 4(-12)(-14) = 576 - 672 = -96 Δ<0\Delta < 0, so roots are complex.


Final Answer:

The two equations with complex roots are:

  • A: y=(x3)2+3y = (x - 3)^2 + 3
  • E: y=12(x+1)22y = -12(x + 1)^2 - 2

Would you like detailed steps for any part? Here are five related questions for further exploration:

  1. What does the discriminant tell us about the nature of roots?
  2. How would you rewrite a quadratic equation in standard form without expanding?
  3. Can you identify real, complex, and repeated roots graphically?
  4. How does the vertex form help in understanding the shape of a parabola?
  5. What are the applications of quadratic equations with complex roots?

Tip: Always check the sign of the discriminant (Δ\Delta) to quickly determine the nature of the roots!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Vertex Form
Standard Form
Discriminant Analysis

Formulas

Discriminant formula: Δ = b^2 - 4ac
Conversion from vertex form to standard form: y = a(x - h)^2 + k → y = ax^2 + bx + c

Theorems

Discriminant Theorem
Properties of Quadratic Equations

Suitable Grade Level

Grades 9-11