#### Question: In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of the marks would have been 210. Find her marks in the two subjects.

#### Video Explanation

#### Explanatory Answer

Step 1: Frame equations from the statements given in the question.

Let her marks in Math be M.

Let her marks in English be E.

Sum of the marks in Math and English is 30 = M + E = 30

∴ M = 30 - E ---------(1)

Had she got 2 more marks in Math, her marks in Math would be (M + 2) ---------(2)

Had she got 3 lesser in English, her marks in English would be (E - 3) ---------(3)

The product of (2) and (3) is 210

i.e., (M + 2) (E - 3) = 210

Substitute M as (30 - E) from (1)

(30 - E + 2)(E - 3) = 210

(32 - E)(E - 3) = 210

32E - 96 – E^{2} + 3E = 210

E^{2} – 35E + 306 = 0

##### Step 2: Find the roots either by factorizing or by using the quadratic formula

Method 1: Factorize the equation to find values of E.

E^{2} - 18E - 17E + 306 = 0

E(E – 18) – 17(E – 18) = 0

(E – 18) (E – 17) = 0

E = 18 and E = 17

If E = 18, M = 30 – E = 30 – 18 = 12

If E = 17, M = 30 – E = 30 – 17 = 13

Method 2: Use the quadratic formula to find values of E.

The roots of ax^{2} + bx + c = 0 are \\frac {-b\pm \sqrt {{b}^{2}-4ac}} {2a})

In this equation, a = 1, b = -35 and c = 306

b^{2} – 4ac = (-35)^{2} – 4 * 1 * 306 = 1225 – 1224 = 1 > 0

Therefore, E = \\frac{-(-35) 士 {\sqrt1}}{2}) = \\frac{35 士 1}{2})

E = 18 and E = 17.

Therefore, M = 12 and M = 13.