Three Ways to Solve a Quadratic Equation

When to take a square root, when to factor out a common term, and the general formula that solves every quadratic once you read off a, b and c correctly.

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A quadratic equation always has two solutions. Which method you reach for depends on the shape of the equation in front of you — and one of the three works every single time.

Concept What makes an equation quadratic
x x² x³ 1 solution 2 solutions 3 solutions the highest power sets the count
In a quadratic the highest power of the variable is 2. That is why it always carries two solutions, even when the two happen to be equal.
Example Method 1 — take the square root

Use this when the equation has the form x^2 = k , with no x term. Solve x^2 = 4 .

Take the square root of both sides.
x = \pm\sqrt{4}
Both signs work: (2)^2 = 4 and (-2)^2 = 4 .
⟹ x = 2 or x = −2

The \pm is essential. Writing only x = 2 loses half the answer.

Example Method 2 — take out a common factor

Use this when every term shares a factor. Solve x^2 - 3x = 0 .

Both terms contain x , so factor it out:
x(x - 3) = 0
If a product is zero, at least one factor must be zero:
x = 0 or x - 3 = 0
⟹ x = 0 or x = 3

Never divide both sides by x here — that would silently discard the solution x = 0 .

Theorem Method 3 — the general formula

For the standard form ax^2 + bx + c = 0 :

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Here a is the coefficient of x^2 , b the coefficient of x , and c the constant. This method solves every quadratic, including those the first two cannot touch.

Example Reading off a, b and c

Arrange each equation from the highest power down, then identify the coefficients.

x^2 = 4 rearranges to x^2 - 4 = 0 : a = 1, b = 0, c = -4
x^2 - 3x = 0 : a = 1, b = -3, c = 0
2x^2 + 4x - 2 = 0 : a = 2, b = 4, c = -2

Watch the signs — a negative coefficient must be carried into the formula as a negative.

Example Applying the formula

Solve 2x^2 + 4x - 2 = 0 , where a = 2, b = 4, c = -2 .

b^2 - 4ac = 16 - 4(2)(-2) = 16 + 16 = 32
x = \frac{-4 \pm \sqrt{32}}{4}
\sqrt{32} = 4\sqrt{2} , so:
x = \frac{-4 \pm 4\sqrt{2}}{4}
⟹ x = −1 + √2 or x = −1 − √2
Summary
  1. A quadratic has highest power 2 and always two solutions.
  2. Square root method: for x² = k, giving x = ±√k.
  3. Factoring: take out the common factor, then set each factor to zero.
  4. General formula: works for every quadratic once written as ax² + bx + c = 0.
  5. Arrange the equation first, mind the signs, and check both answers.
231. Arithmetic Sequences and Series
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