Now we group the terms and factor out the common factors from each pair. From the first two terms, 2x² - 2x, we factor out 2x. From the last two terms, -3x + 3, we factor out -3.

Find the roots of $2x^2 - 5x + 3 = 0$ by factorisation.

Answer: The roots of the equation $2x^2 - 5x + 3 = 0$ are $x = \frac{3}{2}$ and $x = 1$ .

Step-by-step solution

Step 1: Rewrite the middle term

To factorise the quadratic equation $2x^2 - 5x + 3 = 0$ , we need to split the middle term, $-5x$ , into two terms whose product is equal to the product of the first and last terms ( $2x^2 \times 3 = 6x^2$ ) and whose sum is $-5x$ . The two numbers are $-2$ and $-3$ because $(-2) + (-3) = -5$ and $(-2) \times (-3) = 6$ . So, we rewrite $-5x$ as $-2x - 3x$ .

Step 2: Factor by grouping

Now we group the terms and factor out the common factors from each pair. From the first two terms, $2x^2 - 2x$ , we factor out $2x$ . From the last two terms, $-3x + 3$ , we factor out $-3$ .

Step 3: Factor out the common binomial

We can see that $(x - 1)$ is a common factor in both terms. So, we factor out $(x - 1)$ to get the completely factorised form of the quadratic equation.

Step 4: Set each factor to zero

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero to find the possible values of $x$ .

Step 5: Solve for x

We solve each linear equation for $x$ . For the first equation, $2x - 3 = 0$ , we add $3$ to both sides and then divide by $2$ . For the second equation, $x - 1 = 0$ , we add $1$ to both sides.

Step 6: State the roots

The solutions obtained are the roots of the given quadratic equation. These are the values of $x$ that satisfy the original equation.

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Find the roots of 2x2−5x+3=02x^2 - 5x + 3 = 02x2−5x+3=0 by factorisation.