dailymathMainPostsLogin
dailymath logo
dailymath
Login
All practice topics

Quadratic Function

19 practice questions with full worked solutions. Free, no account needed.

Problems & worked solutions

Problem #0008 International

Problem 1 — Quadratic Equations

What is the sum of the solutions to x2−5x+6=0?

Show answer & worked solution
  1. A. −5
  2. B. −1
  3. C. 1
  4. D. 5✓ correct

Factoring gives (x−2)(x−3)=0, so the roots are 2 and 3. Their sum is 5. (Equivalently, −b/a=5/1.)

Problem #0472 International

Problem 2 — Quadratic Function

The discriminant of 2x2−3x−5=0 equals:

Show answer & worked solution
  1. A. 49✓ correct
  2. B. 9
  3. C. −31
  4. D. 29

Δ=(−3)2−4⋅2⋅(−5)=9+40=49.

Problem #0471 International

Problem 3 — Quadratic Function

The roots of x2−5x+6=0 are:

Show answer & worked solution
  1. A. 1 and 6
  2. B. −2 and −3
  3. C. 2 and 3✓ correct
  4. D. −1 and −6

(x−2)(x−3)=0⇒x∈{2,3}.

Problem #0473 International

Problem 4 — Quadratic Function

The vertex (extremum point) of f(x)=x2−4x+3 has x-coordinate:

Show answer & worked solution
  1. A. −4
  2. B. −2
  3. C. 2✓ correct
  4. D. 4

xV=−−42⋅1=2. (And f(2)=4−8+3=−1 is the minimum.)

Problem #0822 International

Problem 5 — Quadratic Equations

Solve by factorising: x2+7x+12=0.

Show answer & worked solution
  1. A. x=−3 or x=−4✓ correct
  2. B. x=3 or x=4
  3. C. x=−2 or x=−6
  4. D. x=−1 or x=−12

Look for two numbers with product 12 and sum 7: these are 3 and 4. x2+7x+12=(x+3)(x+4) (x+3)(x+4)=0 x=−3 or x=−4

Problem #0923 US SAT

Problem 6 — Quadratic Equations

The quadratic equation x2−7x+12=0 has two solutions. What is the sum of its solutions?

Show answer & worked solution
  1. A. −7
  2. B. 7✓ correct
  3. C. 12
  4. D. 1

Factor the quadratic: x2−7x+12=(x−3)(x−4)=0

So the solutions are: x=3orx=4

Their sum is: 3+4=7

This matches Vieta's formula, −b/a=−(−7)/1=7.

Problem #0476 International

Problem 7 — Quadratic Function

For which values of m∈R does x2+(m−2)x+1=0 have a double root?

Show answer & worked solution
  1. A. Only m=0
  2. B. Only m=4
  3. C. m∈{−2,2}
  4. D. m∈{0,4}✓ correct

Δ=(m−2)2−4=0⇒(m−2)2=4⇒m−2=±2⇒m∈{0,4}.

Problem #0478 International

Problem 8 — Quadratic Function

For f:R→R, f(x)=3x2−7x+2, the value of f(2025)⋅f(2) is:

Show answer & worked solution
  1. A. 0✓ correct
  2. B. 1
  3. C. f(2025)
  4. D. A non-zero value depending on 2025

f(2)=12−14+2=0. So f(2025)⋅f(2)=f(2025)⋅0=0, regardless of f(2025).

Problem #0475 International

Problem 9 — Quadratic Function

The solution set of x2−4<0 in R is:

Show answer & worked solution
  1. A. (−∞,−2)∪(2,+∞)
  2. B. {−2,2}
  3. C. (−2,2)✓ correct
  4. D. [−2,2]

(x−2)(x+2)<0⇔x∈(−2,2).

Problem #0477 International

Problem 10 — Quadratic Function

The minimum value of f:R→R, f(x)=x2−6x+8 is:

Show answer & worked solution
  1. A. −1✓ correct
  2. B. 0
  3. C. 1
  4. D. 8

Δ=36−32=4, a=1, so fmin⁡=−44=−1 (attained at x=3).

Which exam are you sitting?