Question

In a class test, the sum of shefali's marks in mathematics and English is 30. Has she got 2 marks more in mathematics and 3 Marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

Answer 1

$\huge \bf \red{SO} \purple{LU} \green{TI} \pink{ON} \blue{:-}$

$\bf \: Let \: Shefali's \: marks \: in \: Mathematics \: be \: \bf \red{x.}$

$\bf \: Then, \: Shefali's \: marks \: in \: English \:$

$\: \: \: \: \: \: \: \: \: \rm = \: ( \bf \red{30 - x})$

$\boxed {\bf \: ∵ The \: sum \: of \: Shefali's \: marks \: in \: Mathematics \: and \: English \: is \: 30}$

$\bf \: According \: to \: the \: question,$

$\: \: \: \: \: \: \: \: \bf(x + 2)(30 - x) - 3 = 210$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf(x + 2)(27 - x) = 210$

$\implies \: \: \: \: \: \: \: \: \: \: \ \: \: \: \: \bf27x - x {}^{2} + 54 - 2x = 210$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf - x {}^{2} + 25 + 54 - 210 = 0$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf - x {}^{2} + 25x - 156 = 0$

$\bf \: Dividing \: by \: - 1$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf \: x {}^{2} - 25x + 156 = 0$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \ \boxed{ \bf{ax {}^{2} + bx {}^{2} + c = 0}}$

$\bf \: which \: is \: a \: quadratic \: equation \: in \: x.$

$\bf \: Here \: \: \: \: \: \: \: \: \: \: \: \: \: \: a = 1 \: b = - 25 \: c = 156$

$\bf \: So \: \: \: \: \: \: \: \: \: \: \: \: \: d = b {}^{2} - 4ac$

$\bf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = ( - 25) {}^{2} - 4(1)(156)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf = \: 625 - 624 = 1$

$\bf \: Using \: the \: quadratic \: formula,$

$\bf \: we \: get$

$\bf \: x = \frac{ - b \: ± \: \sqrt{D} }{2a} = \frac{25 \: ± \: \sqrt{1} }{2(1)} = \frac{25 \: ± \: 1}{2}$

$\implies \: \: \: \: \: \: \: \bf \: x = \frac{25 + 1}{2} \frac{25 - 1}{2} = 13 \: 12$

$\implies \: \: \: \: \: \: \bf \: 30 - x = 30 - 13 \: or \: 30 - 12$

$\implies \: \: \: \bf \: 30 - x = 17 \: 18$

$\bf \: Therefore, \: either \: Shefali's \: marks \: in \: Mathematics \: are \\ \bf \red{13} \: and \: in \: English \: \red{ 17 }\: or \: her \: marks \: in \: Mathematics \: are \\ \bf \red{ 12 }\: and \: in \: English \: \red{ 18}.$

Answer 2

$\huge\boxed{\fcolorbox{yellow}{black}{❥ANSWER}}$

Let us say, the marks of Shefali in Maths be x.

Then, the marks in English will be 30 – x.

As per the given question,

(x + 2)(30 – x – 3) = 210

(x + 2)(27 – x) = 210

⇒ -x2 + 25x + 54 = 210

⇒ x2 – 25x + 156 = 0

⇒ x2 – 12x – 13x + 156 = 0

⇒ x(x – 12) -13(x – 12) = 0

⇒ (x – 12)(x – 13) = 0

⇒ x = 12, 13

Therefore, if the marks in Maths are 12, then marks in English will be 30 – 12 = 18 and the marks in Maths are 13, then marks in English will be 30 – 13 = 17