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 their marks would have been $210$. Find her marks in the two subjects.

Answer 1

Step $1:$ Obtain a quadratic equation.

Assume that, Shefali got $x$ marks in Mathematics.

Since, the sum of Shefali’s marks in Mathematics and English is $30$.

Therefore, Shefali got $(30-x)$ marks in English.

So, according to the question.

$$\begin{gathered} (x+2)(30-x-3)=210 \\ \Rightarrow (x+2)(27-x)=210 \\ \Rightarrow 27x-x^2+54-2x=210 \\ \Rightarrow -x^2+25x-156=0 \\ \Rightarrow x^2-25x+156=0 \end{gathered}$$

Step $2:$ Compute the required values.

We know that, if a quadratic equation is in the form $a{x}^2+bx+c=0$, then according to the quadratic formula $\mathbf{x}\mathbf{=}\frac{\mathbf{-}\mathbf{b}\mathbf{\pm }\sqrt{{\mathbf{b}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{ac}}}{\mathbf{2}\mathbf{a}}$.

Use the quadratic formula to solve the obtained quadratic equation.

$$\begin{gathered} x=\frac{25\pm \sqrt{625-624}}{2} \\ \Rightarrow x=\frac{25\pm 1}{2} \\ \Rightarrow x=\left\{\frac{24}{2},\frac{26}{2}\right\} \\ \Rightarrow x=\left\{12,13\right\} \end{gathered}$$

Therefore, the values of $x$ are $12$ and $13$.

So, when Shefali got $12$ marks in Mathematics.

Then her marks in English will be $30-x=18$.

And when Shefali got $13$ marks in Mathematics.

Then her marks in English will be $30-x=17$.

Hence, Shefali's marks in two subjects are as: $12$ in mathematics and $18$ in English or $13$ in Mathematics and $17$ in English.