Algebra

x^2 - 5x + 6 = 0

Step-by-step solution with explanation

Final Answer

and

Step-by-step solution

1

Write down the equation clearly

We start by looking at the equation. It is a quadratic equation, meaning the highest power of x is 2. Quadratics can have up to 2 solutions.
2

Find two numbers that factor nicely

To factor, we look for two numbers where the product equals the constant term (6) and the sum equals the middle coefficient (-5). Think: what pairs multiply to 6? Try -2 and -3: they multiply to 6 and add to -5. ✓
3

Write the factored form

We rewrite the equation as two binomials multiplied together. You can check by expanding: . ✓
4

Apply the Zero Product Property

If two things multiply to zero, at least one of them must be zero. So we set each factor equal to zero and solve each mini-equation separately.
5

Solve each equation for x

Adding 2 to both sides of the first equation gives x = 2. Adding 3 to both sides of the second gives x = 3. These are both solutions.
6

Check both answers in original equation

Plug each answer back into the original equation. Both give 0, so both solutions are confirmed correct.

Understanding this problem

Learning Insight

Factoring works because of the Zero Product Property: if A × B = 0, then A = 0 or B = 0. This is a unique property of zero — no other number has it. That's why we always move everything to one side and set the equation equal to zero before factoring.

Quick Tip

For , just ask: 'What two numbers multiply to c AND add to b?' If you find them, you can factor instantly without any formula.

Common Mistake

Students often stop after factoring and write the answer as x = -2 and x = -3, forgetting to flip the signs. If the factor is , then x = +2, not -2.