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Exercises on quadratic equations with the formula

Solve quadratic equations (ax² + bx + c = 0) using the quadratic formula, including incomplete ones. Theory, discriminant, and step-by-step solved exercises, with instantly corrected practice.

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Quadratic equations: quick theory

A quadratic equation has the form ax2 + bx + c = 0, with a ≠ 0. It is solved using the quadratic formula:

x = ( −b ± √(b2 − 4ac) ) / 2a

The discriminant Δ = b2 − 4ac indicates how many solutions there are: if Δ > 0 there are two, if Δ = 0 there is one (double), and if Δ < 0 there are no real solutions.

Incomplete ones (when b or c is missing) are solved faster without the formula: by factoring or by isolating x2.

Step-by-step solved exercises on quadratic equations

Solved examples to show you the method. Then, practice with instant correction.

Exercise 1Solve x² − 5x + 6 = 0.

Step-by-step solution

  1. We identify a = 1, b = −5, c = 6.
  2. Discriminant: Δ = (−5)² − 4·1·6 = 25 − 24 = 1.
  3. We apply the formula: x = (5 ± √1) / 2 = (5 ± 1) / 2.
  4. The two solutions: x = 6/2 = 3 and x = 4/2 = 2.
Solutions: x = 3 and x = 2.

Exercise 2Solve the incomplete equation x² − 4 = 0.

Step-by-step solution

  1. We isolate x²: x² = 4.
  2. We take the square root (with both signs): x = ±√4.
Solutions: x = 2 and x = −2.

Exercise 3Solve 2x² + 3x − 2 = 0.

Step-by-step solution

  1. a = 2, b = 3, c = −2.
  2. Discriminant: Δ = 3² − 4·2·(−2) = 9 + 16 = 25, and √25 = 5.
  3. Formula: x = (−3 ± 5) / (2·2) = (−3 ± 5) / 4.
  4. Solutions: x = 2/4 = 1/2 and x = −8/4 = −2.
Solutions: x = 1/2 and x = −2.

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This topic is studied in:

3rd year of ESO4th year of ESO1st year of Bachillerato

Frequently asked questions about quadratic equations

What is the quadratic formula?

For ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. The ± sign gives the two possible solutions.

What is the discriminant?

It is Δ = b² − 4ac. If it is positive there are two real solutions, if it is zero there is a double solution, and if it is negative there are no real solutions.

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