The quadratic formula is the one method that solves every quadratic. DeltaMath itself started in 2009 as a single practice program for substituting into this formula, because its creator noticed students consistently made two mistakes: not setting the equation to zero, and mis-computing the discriminant when b is negative.
Those two mistakes are still the ones to watch. Everything else is arithmetic and radical simplification.
How to solve it, step by step
Set the equation equal to zero
Move every term to one side so it reads ax² + bx + c = 0. For 2x² = 5x + 3, rewrite as 2x² − 5x − 3 = 0.
Identify a, b, and c with their signs
a = 2, b = −5, c = −3. Write them down; do not hold them in your head.
Compute the discriminant b² − 4ac
(−5)² − 4(2)(−3) = 25 + 24 = 49. Note that (−5)² is +25 and −4ac becomes positive because c is negative.
Substitute into the formula
x = (−(−5) ± √49) / (2·2) = (5 ± 7) / 4.
Simplify both solutions
x = 12/4 = 3 or x = −2/4 = −1/2. If the radical does not simplify to an integer, leave it in simplest radical form.
Worked example
Solve x² − 6x + 2 = 0.
- a = 1, b = −6, c = 2
- Discriminant: (−6)² − 4(1)(2) = 36 − 8 = 28
- x = (6 ± √28) / 2
- √28 = √(4·7) = 2√7
- x = (6 ± 2√7) / 2 = 3 ± √7
Answer x = 3 + √7 or x = 3 − √7
Solve x² + 4x + 13 = 0 (Algebra 2, complex solutions).
- Discriminant: 16 − 52 = −36
- x = (−4 ± √(−36)) / 2 = (−4 ± 6i) / 2
Answer x = −2 ± 3i
Mistakes DeltaMath marks wrong
- Using the formula before setting the equation equal to zero.
- Computing (−5)² as −25.
- Forgetting the ± and reporting only one solution.
- Dividing only the radical term by 2a: (6 ± 2√7)/2 is 3 ± √7, not 6 ± √7.
- Leaving √28 unsimplified when exact simplest form is required.
How to type the answer
- Type radicals with sqrt: 3+sqrt(7) and 3-sqrt(7), or use the math keyboard's √ key.
- Separate two answers with a comma if the box asks for all solutions.
- For complex answers write a+bi form, for example -2+3i.