Find the roots of the equation 2x2 - 5x + 3 = 0 by factorisation.
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Step-by-Step Solution
Step 1: Identify coefficients and split the middle term
To factorize a quadratic equation of the form ax2+bx+c=0, we need to find two numbers whose product is ac and whose sum is b. Here, a=2, b=−5, and c=3. So, we need two numbers whose product is 2×3=6 and whose sum is −5. These numbers are −2 and −3. We then split the middle term, −5x, into -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, 2x2−2x, we factor out 2x, leaving x−1. From the last two terms, −3x+3, we factor out −3, also leaving x−1.
Step 3: Factor the common binomial
Since (x−1) is a common factor in both terms, we can factor it out, which gives us the product of two binomials: (x−1)(2x−3)=0.
Step 4: Solve for x
For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x. This gives us the two roots of the quadratic equation.