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Find the roots of the quadratic equation 2x┬▓ - 7x + 3 = 0.
3, 0.5
2, 1
3, 1
1.5, 0.5
3, 0.5
Use the sum and product rule: product of roots is c/a=3/2=1.5, and sum of roots is тИТb/a=7/2=3.5. Only option A satisfies both product (3imes0.5=1.5) and sum (3+0.5=3.5).
A quadratic equation in the standard form 2x┬▓ - 7x + 3 = 0 where a=2, b=-7, and c=3.
x=2aтИТb┬▒b2тИТ4acтАЛтАЛ
Use the sum and product rule: product of roots is c/a=3/2=1.5, and sum of roots is тИТb/a=7/2=3.5. Only option A satisfies both product (3imes0.5=1.5) and sum (3+0.5=3.5).
Students often fail to divide the final factors by the leading coefficient a=2, leading to an incorrect result of 6 and 1 instead of 3 and 0.5.
Identify coefficients
Identify the values of a, b, and c from the equation 2x2тИТ7x+3=0.
a=2,b=тИТ7,c=3
Split the middle term
Find two numbers whose sum is тИТ7 and whose product is 2imes3=6. These numbers are тИТ6 and тИТ1. Rewrite the equation as 2x2тИТ6xтИТx+3=0.
2x(xтИТ3)тИТ1(xтИТ3)=0
Factorize the expression
Factor out the common binomial (xтИТ3) to get (2xтИТ1)(xтИТ3)=0.
(2xтИТ1)(xтИТ3)=0
Solve for x
Set each factor to zero to find the roots: 2xтИТ1=0тЯ╣x=0.5 and xтИТ3=0тЯ╣x=3.
x=0.5,x=3
A is correct because the roots calculated by factorizing the quadratic equation 2x2тИТ7x+3=0 are 3 and 0.5.
This splitting middle-term method is essentially the reverse of the Vieta's formulas, which is a vital shortcut for checking roots of higher-order polynomials in competitive exams.
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