Worksheet: Roots of Quadratic Functions

In this worksheet, we will practice solving quadratic equations by factoring.


Given that 𝑓(π‘₯) is a quadratic function and π‘₯=4 is a root of the equation 𝑓(π‘₯)=0, what is the value of 𝑓(4)?


Find the solution set of 5𝑦+24π‘¦βˆ’5=0 in ℝ.

  • A{1,βˆ’5}
  • B{βˆ’1,5}
  • C{5,βˆ’5}
  • Dο¬βˆ’15,5
  • E15,βˆ’5


What are the zeros of 𝑓(π‘₯)=(π‘₯+πœ‹)βˆ’π‘’οŠ¨?

  • Aβˆ’πœ‹+βˆšπ‘’ and βˆ’πœ‹βˆ’βˆšπ‘’
  • Bβˆšπ‘’βˆ’πœ‹ and βˆšπ‘’+πœ‹
  • Cβˆ’πœ‹βˆ’βˆšπ‘’ and βˆ’πœ‹βˆ’βˆšπ‘’
  • Dβˆ’πœ‹+𝑒 and βˆ’πœ‹βˆ’π‘’
  • Eπ‘’βˆ’πœ‹ and 𝑒+πœ‹


Find the set of zeros of the function 𝑓(π‘₯)=(π‘₯βˆ’2)(π‘₯+5)βˆ’18.

  • A{2,βˆ’5}
  • B{βˆ’7,4}
  • C{βˆ’7,βˆ’4}
  • D{7,βˆ’4}
  • E{7,4}


Which of the following is a solution for π‘Ž given the set of zeros of the function 𝑓(π‘₯)=π‘₯+π‘ŽοŠ¨ is empty?

  • Aβˆ’44
  • B44
  • C0
  • Dβˆ’13


If a parabola intersects the π‘₯-axis at two points, find the number of roots for the equation in ℝ.

  • Aone
  • Bzero
  • Ctwo
  • Dfour
  • Ethree


If a parabola touches the π‘₯-axis at a single point, determine the number of roots in ℝ.

  • Aone
  • Btwo
  • Cthree
  • Dzero
  • Efour


If a parabola does not intersect the π‘₯-axis, determine the number of roots in ℝ.

  • Athree
  • Bfour
  • Cone
  • Dzero
  • Etwo


Given the curve of the quadratic function 𝑓 does not intersect the π‘₯-axis, determine 𝑍(𝑓). Recall that 𝑍(𝑓) is the set of zeros of the function 𝑓.

  • Aβˆ…
  • Bβ„š
  • Cβ„•
  • D{0}
  • Eℝ


The following expressions are equivalent ways of writing the function 𝑓.

  1. 𝑓(π‘₯)=π‘₯+8π‘₯+15
  2. 𝑓(π‘₯)=(π‘₯+4)βˆ’1
  3. 𝑓(π‘₯)=(π‘₯+5)(π‘₯+3)

Use expression 1 to find the value of 𝑓 when π‘₯=0.

Identify the minimum value of 𝑓 using expression 2.

Identify the zeros of 𝑓 using expression 3.

  • Aβˆ’5, 3
  • B5, 3
  • C8, 15
  • Dβˆ’5, βˆ’3
  • E5, βˆ’3


Find the values of π‘Ž and 𝑏 given the set {βˆ’4,2} contains the zeros of the function 𝑓(π‘₯)=π‘Žπ‘₯+𝑏π‘₯βˆ’32.

  • Aπ‘Ž=4, 𝑏=8
  • Bπ‘Ž=βˆ’4, 𝑏=βˆ’8
  • Cπ‘Ž=8, 𝑏=4
  • Dπ‘Ž=βˆ’8, 𝑏=βˆ’4


Find the value of π‘Ž, given the set {βˆ’9} contains the zero of the function 𝑓(π‘₯)=π‘₯βˆ’2π‘Žπ‘₯+π‘ŽοŠ¨οŠ¨.


Given that 𝑓(π‘₯)=π‘₯βˆ’2π‘₯βˆ’8 and 𝑓(4)=0, find the other root of 𝑓(π‘₯).

  • Aπ‘₯=1
  • Bπ‘₯=βˆ’1
  • Cπ‘₯=βˆ’4
  • Dπ‘₯=2
  • Eπ‘₯=βˆ’2


Determine the points at which the graph of the function π‘“βˆΆπ‘“(π‘₯)=π‘₯βˆ’π‘₯ intersects the π‘₯-axis.

  • A(2,0), (0,1)
  • B(2,0), (βˆ’1,0)
  • C(0,0), (1,0)
  • D(0,0), (βˆ’1,0)


Find the set of zeros of the function 𝑓(π‘₯)=1βˆ’9π‘₯.

  • Aο¬βˆ’13
  • B{3}
  • C{3,βˆ’3}
  • D13
  • E13,βˆ’13


If the graph of the quadratic function, 𝑓(π‘₯), does not intersect the π‘₯-axis, how many solutions are there to the equation 𝑓(π‘₯)=0?

  • Azero
  • Bone
  • Can infinite number
  • Dtwo


At which values of π‘₯ does the graph of 𝑦=12π‘₯βˆ’8π‘₯ cross the π‘₯-axis?

  • A0 and 23
  • B0 and βˆ’83
  • C0 and βˆ’23
  • D0 and 83
  • E0 and 2

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