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Mathematics

Grade10

Easy

Question

The sum of the roots of a quadratic equation is equal to 7 and the difference of the roots is equal to 5. Which of these could be the equation?

$x^{2} - 7 x + 6 = 0$
$x^{2} + 7 x + 6 = 0$
$x^{2} - 6 x + 8 = 0$
$x^{2} + 7 x + 5 = 0$

The correct answer is: $x squared minus 7 x plus 6 equals 0$

Let, the equation be $a x squared plus b x plus c equals 0$ and it can be written as $x squared plus b over a x plus c over a equals 0.$ .....(1)
Then , the sum of roots = $negative b over a$ and product of roots = $c over a$ .
Let, the roots be $alpha space comma space beta$ .
Then , $alpha plus beta equals negative b over a space space space comma space space space space alpha beta equals c over a$ .
Therefore, we have $alpha plus beta equals 7 space....... space left parenthesis 1 right parenthesis alpha minus beta equals 5 space....... space left parenthesis 2 right parenthesis$
Solving (1) and (2), we get
$space space alpha plus beta equals space 7 space space alpha minus beta equals space 5 minus negative negative negative negative negative space space 2 alpha space space space space space equals space 12 rightwards double arrow alpha equals 6 space space space beta equals 1$
Now we get, $alpha plus beta equals 7 space rightwards double arrow negative b over a equals 7 space rightwards double arrow b over a equals negative 7$ and $alpha beta equals 6 space rightwards double arrow c over a equals 6$ .
Now,putting the value of $b over a space comma space c over a$ in the first equation , we get the required equation which is $x squared minus 7 x plus 6 equals 0$ .

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