How to Solve 2x² − 4x − 6 = 0

Look for a common factor first

Before anything else, check whether all three coefficients share a factor. Here 2, −4 and −6 are all divisible by 2.

Dividing the whole equation by 2 gives x² − 2x − 3 = 0.

This is allowed because the right-hand side is zero, and zero divided by 2 is still zero. The roots are completely unchanged.

This step is skipped constantly, and it turns an awkward-looking problem into an easy one. Whenever a is not 1, check for a common factor before deciding the equation is difficult.

Now factor the simpler equation

For x² − 2x − 3 = 0, find two numbers that multiply to −3 and add to −2.

The pairs multiplying to −3 are 1 and −3, or −1 and 3. Of those, 1 and −3 add to −2.

So it factors as (x + 1)(x − 3) = 0, giving x = −1 or x = 3.

Or use the formula on the original

If you did not spot the common factor, the formula handles the original equation without complaint. With a = 2, b = −4, c = −6:

Discriminant: (−4)² − 4(2)(−6) = 16 + 48 = 64.

√64 = 8, and 2a = 4.

Plus branch: x = (4 + 8) ÷ 4 = 12 ÷ 4 = 3.

Minus branch: x = (4 − 8) ÷ 4 = −4 ÷ 4 = −1.

Same answers, as they must be. The calculator on this page is loaded with the original equation so you can follow the substitution.

Factoring directly, without simplifying

It is possible to factor 2x² − 4x − 6 as it stands, and it is worth seeing once.

You need two numbers multiplying to a × c = 2 × −6 = −12, and adding to b = −4. Those are −6 and 2.

Split the middle term: 2x² − 6x + 2x − 6.

Group in pairs: 2x(x − 3) + 2(x − 3).

Factor out the common bracket: (2x + 2)(x − 3), which is 2(x + 1)(x − 3).

Setting each factor to zero gives the same roots, −1 and 3. It works, but it is clearly more effort than dividing by 2 at the start.

Checking

For x = 3: 2(9) − 4(3) − 6 = 18 − 12 − 6 = 0. Correct.

For x = −1: 2(1) − 4(−1) − 6 = 2 + 4 − 6 = 0. Correct.

A Vieta check as well: the roots sum to 3 + (−1) = 2, and −b ÷ a = 4 ÷ 2 = 2. They multiply to −3, and c ÷ a = −6 ÷ 2 = −3. Both agree.