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The Quadratic Formula, Explained

The quadratic formula solves every quadratic equation, including the ones that do not factor. Reading the discriminant first tells you what kind of answer to expect, which is how you catch an arithmetic slip before it becomes a wrong answer.

The formula

For any equation of the form ax^2 + bx + c = 0, the solutions are x = (-b plus or minus the square root of (b^2 - 4ac)) all divided by 2a. Write the equation in that standard form first, with everything on one side and zero on the other.

Read the discriminant first

  • b^2 - 4ac greater than zero: two different real solutions.
  • b^2 - 4ac equal to zero: one repeated real solution.
  • b^2 - 4ac less than zero: no real solutions.
  • A perfect square discriminant means the quadratic would also have factored.

Identifying a, b, and c

Signs belong to the coefficients. In x^2 - 4x + 1, b is -4, not 4. Writing a, b, and c down explicitly before substituting is the habit that prevents most errors with this formula.

Worked examples

Example 1

Question

Solve: 2x^2 + 3x - 5 = 0

Answer

x = 1 or x = -5/2

Steps

  1. 1Identify: a = 2, b = 3, c = -5.
  2. 2Discriminant: 3^2 - 4(2)(-5) = 9 + 40 = 49, a perfect square, so expect two neat answers.
  3. 3Square root of 49 is 7.
  4. 4Substitute: x = (-3 plus or minus 7) / (2 x 2) = (-3 plus or minus 7) / 4.
  5. 5First root: (-3 + 7)/4 = 4/4 = 1.
  6. 6Second root: (-3 - 7)/4 = -10/4 = -5/2.

A perfect square discriminant means this one factors as (2x + 5)(x - 1) = 0. Both routes give the same answers.

Example 2

Question

Solve: x^2 - 4x + 1 = 0

Answer

x = 2 + root 3 or x = 2 - root 3

Steps

  1. 1Identify: a = 1, b = -4, c = 1. Note that b is negative.
  2. 2Discriminant: (-4)^2 - 4(1)(1) = 16 - 4 = 12. Positive but not a perfect square, so the roots are irrational.
  3. 3Substitute: x = (4 plus or minus root 12) / 2.
  4. 4Simplify the surd: root 12 = 2 root 3.
  5. 5So x = (4 plus or minus 2 root 3) / 2 = 2 plus or minus root 3.

Dividing every term by 2 at the end is required. Cancelling only the 4 and leaving root 12 over 2 is a common half-simplification.

Example 3

Question

Solve: x^2 + 2x + 5 = 0

Answer

No real solutions

Steps

  1. 1Identify: a = 1, b = 2, c = 5.
  2. 2Discriminant: 2^2 - 4(1)(5) = 4 - 20 = -16.
  3. 3The discriminant is negative, so the square root has no real value.
  4. 4Therefore the equation has no real solutions, and the parabola never crosses the x-axis.
  5. 5If your course covers complex numbers, the solutions are x = -1 plus or minus 2i.

Common mistakes

Dropping the sign of b: using b = 4 for x^2 - 4x + 1.

b is -4. Write a, b, and c out with their signs before substituting anything.

Dividing only part of the numerator by 2a.

The whole numerator is divided. Keep it over one fraction bar until the last step.

Using the formula before rearranging to standard form.

Everything on one side, zero on the other, then identify a, b, and c.

Reporting a negative discriminant as an error.

It is an answer. No real solutions is a complete response unless the course expects complex roots.

Practice it

Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.

  • 1. Solve: x^2 - 7x + 10 = 0

    Show answer

    Discriminant 9, so x = (7 plus or minus 3)/2, giving x = 5 or x = 2.

  • 2. Solve: 3x^2 + 2x - 1 = 0

    Show answer

    Discriminant 16, so x = (-2 plus or minus 4)/6, giving x = 1/3 or x = -1.

  • 3. Solve: 4x^2 - 4x + 1 = 0

    Show answer

    Discriminant 0, so one repeated root: x = 1/2.

Questions people ask

When should I use the formula instead of factoring?
Whenever the discriminant is not a perfect square, or whenever you have spent more than about twenty seconds hunting for factors.
Does the formula always work?
For every quadratic, yes. Factoring only works for some. That reliability is why the formula is worth memorising.
What does the discriminant mean graphically?
It counts how many times the parabola crosses the x-axis: twice, once as a touch, or not at all.
Should I leave surds or convert to decimals?
Leave the exact form unless the question asks for a decimal. 2 plus root 3 is exact; 3.73 is rounded.
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