Equations and inequalities

Quadratic equations

Quadratic equations are a core topic of the T-Kurs entrance test at München and many other Kollegs. German schools use the pq formula, so learn it in that form.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• Bring the equation to the form x² + px + q = 0 (divide by the leading coefficient first).
• pq formula: x = −p/2 ± √((p/2)² − q).
• The discriminant (p/2)² − q decides: positive gives two solutions, zero one, negative none.
• If there is no constant term, factor out x: x² − 5x = x(x − 5) = 0.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Solve x² + 7x + 10 = 0. Give the larger root.

   • a-7
   • b-5
   • c-2
   • d2
   Show solution

   Answer: -2

   The roots r₁, r₂ of x² + bx + c satisfy r₁·r₂ = c and r₁ + r₂ = −b.

   Two numbers with product 10 and sum -7: -5 and -2. Factor (x + 5)(x + 2) = 0.

2. 2. Solve x² − 4 = 0. Give the smaller root.

   • a2
   • b0
   • c-1
   • d-2
   Show solution

   Answer: -2

   The roots r₁, r₂ of x² + bx + c satisfy r₁·r₂ = c and r₁ + r₂ = −b.

   Two numbers with product -4 and sum 0: -2 and 2. Factor (x + 2)(x − 2) = 0.

3. 3. For which positive k does x² + kx + 49 = 0 have exactly one solution?

   • a49
   • b28
   • c7
   • d14
   Show solution

   Answer: 14

   Exactly one solution when the discriminant b² − 4ac is 0.

   k² − 4·49 = 0 ⇒ k² = 196 ⇒ k = 14.

4. 4. Solve x² − 11x + 24 = 0. Give the larger root.

   • a11
   • b3
   • c8
   • d-8
   Show solution

   Answer: 8

   The roots r₁, r₂ of x² + bx + c satisfy r₁·r₂ = c and r₁ + r₂ = −b.

   Two numbers with product 24 and sum 11: 3 and 8. Factor (x − 3)(x − 8) = 0.

5. 5. Solve x² − 2x − 3 = 0. Give the larger root.

   • a2
   • b3
   • c-3
   • d-1
   Show solution

   Answer: 3

   The roots r₁, r₂ of x² + bx + c satisfy r₁·r₂ = c and r₁ + r₂ = −b.

   Two numbers with product -3 and sum 2: -1 and 3. Factor (x + 1)(x − 3) = 0.

6. 6. Solve x² + 7x + 12 = 0. Give the larger root.

   • a-7
   • b-3
   • c3
   • d-4
   Show solution

   Answer: -3

   The roots r₁, r₂ of x² + bx + c satisfy r₁·r₂ = c and r₁ + r₂ = −b.

   Two numbers with product 12 and sum -7: -4 and -3. Factor (x + 4)(x + 3) = 0.

Common mistakes

• Using the pq formula without first dividing by the coefficient of x².
• Dividing by x and losing the solution x = 0.
• Sign errors with −p/2 when p is negative.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs