We are given a quadratic equation in the standard form ax² + bx + c = 0. We need to identify the coefficients a, b, and c from the given equation 2x² - 5x + 3 = 0.

To factorise the quadratic equation by splitting the middle term, we first multiply the coefficient of x² (which is a) and the constant term (which is c). In this case, a=2 and c=3.

Group the terms and factor out the common monomial from each pair. From 2x² - 2x, we factor out 2x, leaving x-1. From -3x + 3, we factor out -3, also leaving x-1.

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

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

Step-by-step solution

Step 1: Identify coefficients

We are given a quadratic equation in the standard form $ax^2 + bx + c = 0$ . We need to identify the coefficients $a$ , $b$ , and $c$ from the given equation $2x^2 - 5x + 3 = 0$ .

Step 2: Multiply 'a' and 'c'

To factorise the quadratic equation by splitting the middle term, we first multiply the coefficient of $x^2$ (which is $a$ ) and the constant term (which is $c$ ). In this case, $a=2$ and $c=3$ .

Step 3: Find two numbers whose product is 'ac' and sum is 'b'

We need to find two numbers, let's call them $p$ and $q$ , such that their product is $ac=6$ and their sum is $b=-5$ . By trying different factors of 6, we find that $-2$ and $-3$ satisfy these conditions.

Step 4: Split the middle term

Now we split the middle term, $-5x$ , into two terms using the numbers we found: $-2x$ and $-3x$ . This allows us to group terms for factorization.

Step 5: Factor by grouping

Group the terms and factor out the common monomial from each pair. From $2x^2 - 2x$ , we factor out $2x$ , leaving $x-1$ . From $-3x + 3$ , we factor out $-3$ , also leaving $x-1$ .

Step 6: Factor out the common binomial

We can see that $(x - 1)$ is a common binomial factor in both terms. Factor $(x - 1)$ out to express the quadratic equation as a product of two linear factors.

Step 7: Set each factor to zero

According to the property of real numbers, if the product of two factors is zero, then at least one of the factors must be zero. So, we set each linear factor equal to zero to find the roots (Ref. NCERT Class 10 Mathematics - Quadratic Equations, Page 1).

Step 8: Solve for x

Solve each linear equation for $x$ . The first equation gives $x=1$ . The second equation gives $2x=3$ , which means $x=\frac{3}{2}$ . These are the two roots of the quadratic equation.

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