**Quadratic Equations Class ten NCERT Solutions**

**Chapter 4: Quadratic Equations**

**TEXTBOOK’S EXERCISE 4.3.**

**Find the roots of the following equations :**

⇒ x^{2} – 3x – 1 = 0

This is of the form ax^{2} + bx +
c = 0.

Where a = 1, b = – 3, c = –1

Now, b^{2} – 4ac = (–3)^{2} – 4 × 1 × (–1) = 9 + 4 = 13

are the roots of given quadratic equation.

⇒ – 11 × 30 = 11(x^{2} – 3x – 28)

⇒ –30 = x^{2}
– 3x – 28

⇒ x^{2}
– 3x – 28 + 30 = 0

⇒ x^{2}
– 3x + 2 = 0

This is of the form ax^{2}
+ bx + c = 0.

Where, a = 1, b = –3, c = 2

Now, b^{2} – 4ac = (–3)2 – 4 × 1 × 2 = 9 – 8 = 1

So, roots are

or 4/2 and 2/2 or 2 and 1

Hence, 2 and 1 are the roots of the given quadratic equation.

**4. The sum of the reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is 1/3. Find the present age. **

**Sol.
**Let Rehman’s present age = x
years

3 years ago Rehman’s age = (x – 3) years

5 years from now Rehman’s age = (x + 5) years

As per condition:

⇒ 6x +
6 = x^{2} + 2x –
15

⇒ x^{2} + 2x –
15 – 6x – 6 = 0

⇒ x^{2} – 4x –
21 = 0, which is a quadratic equation in x.

This is of the form ax^{2} + bx + c = 0

Where, a = 1, b = – 4, c = –21

Now, b^{2} – 4ac = (– 4)^{2} – 4 × 1 × (–21)

= 16 + 84 = 100

So, roots are 4+ 10/ 2 and 4 – 10/ 2 or 7 and –3

As age cannot be negative.

∴ x = 7

Hence, Rehman’s present age = 7 years.

**5. 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. **

**Sol.
**Let Shefali got x marks
in Mathematics.

∴ Shefali’s marks in English = 30 – x

As per condition :

Shefali’s marks in Mathematics = x + 2

Shefali’s marks in English = 27 – x

∴ Required product = (x + 2) (27 – x)

= 27x – x^{2} + 54 – 2x

= –x^{2} + 25x + 54

According to the given problem,

–x^{2} + 25x + 54 = 210

⇒ –x^{2} +
25x + 54 – 210 = 0

⇒ –x^{2} +
25x – 156 = 0

⇒ x^{2} –
25x + 156 = 0

⇒ x^{2} –
12x – 13x + 156 = 0

⇒ (x – 12) (x – 13) = 0 ⇒ x = 12 or x = 13

**Case
I : **When x = 13, then

Shefali’s marks in Mathematics = 13

Shefali’s marks in English = 30 – 13 = 17

**Case
II : **When x = 12, then

Shefali’s marks in Mathematics = 12

Shefali’s marks in English = 30 – 12 = 18 Hence, Shefali’s marks in the two subjects are 13 and 17 or 12 and 18.