Quadratic Equations Class ten NCERT Solutions
Chapter 4: Quadratic Equations
TEXTBOOK’S EXERCISE 4.3.
Find the roots of the following equations :
⇒ x2 – 3x – 1 = 0
This is of the form ax2 + bx + c = 0.
Where a = 1, b = – 3, c = –1
Now, b2 – 4ac = (–3)2 – 4 × 1 × (–1) = 9 + 4 = 13
are the roots of given quadratic equation.
⇒ – 11 × 30 = 11(x2 – 3x – 28)
⇒ –30 = x2 – 3x – 28
⇒ x2 – 3x – 28 + 30 = 0
⇒ x2 – 3x + 2 = 0
This is of the form ax2 + bx + c = 0.
Where, a = 1, b = –3, c = 2
Now, b2 – 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 = x2 + 2x – 15
⇒ x2 + 2x – 15 – 6x – 6 = 0
⇒ x2 – 4x – 21 = 0, which is a quadratic equation in x.
This is of the form ax2 + bx + c = 0
Where, a = 1, b = – 4, c = –21
Now, b2 – 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 – x2 + 54 – 2x
= –x2 + 25x + 54
According to the given problem,
–x2 + 25x + 54 = 210
⇒ –x2 + 25x + 54 – 210 = 0
⇒ –x2 + 25x – 156 = 0
⇒ x2 – 25x + 156 = 0
⇒ x2 – 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.