We group the first 2 terms and the last 2 terms. We factor out x from x² - 3x to get x(x - 3), and -2 from -2x + 6 to get -2(x - 3).

$x^2 - 5x + 6 = 0$ $x^2 - 3x - 2x + 3 \times 2 = 0$ $x(x - 3) - 2(x - 3) = 0$ $(x - 3)(x - 2) = 0$ $x - 3 = 0 \quad x - 2 = 0$ $x = 3 \quad x = 2$ $x = 3, 2$

Answer: $x = 2, 3$

Step-by-step solution

Step 1: Identify coefficients and split the middle term

For the quadratic equation $x^2 - 5x + 6 = 0$ , we find 2 numbers whose product is 6 and whose sum is $-5$ . These numbers are $-3$ and $-2$ , so we split $-5x$ into - $3x - 2x$ .

Step 2: Factor by grouping

We group the first 2 terms and the last 2 terms. We factor out $x$ from $x^2 - 3x$ to get x(x - 3), and $-2$ from $-2x + 6$ to get $-2(x - 3)$ .

Step 3: Factor out the common binomial

Both grouped terms share the common factor $(x - 3)$ . Factoring it out leaves the product $(x - 3)(x - 2) = 0$ .

Step 4: Apply zero product property to find roots

By the zero product property, setting each linear factor equal to 0 gives $x - 3 = 0$ or $x - 2 = 0$ . Solving each linear equation gives $x = 3$ or $x = 2$ .

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