Quadratic Equations Chapter 4 Class 10 Maths NCERT Textbook With Solutions PDF

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NCERT Class 10 Maths Textbook Chapter 4 With Answer Book PDF Free Download

Quadratic Equations

Chapter 4: Quadratic Equations

4.4 Solution of a Quadratic Equation by Completing the Square

In the previous section, you have learned one method of obtaining the roots of a quadratic equation. In this section, we shall study another method.

Consider the following situation:
The product of Sunita’s age (in years) two years ago and her age four years from now is one more than twice her present age.

What is her present age? To answer this, let her present age (in years) be x. Then the product of her ages two years ago and four years from now is (x – 2)(x + 4)

AuthorNCERT
Language English
No. of Pages23
PDF Size1783 KB
CategoryMathematics
Source/ Creditsncert.nic.in

NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations

1. Check whether the following are quadratic equations:

(i) (x + 1)2 = 2(x – 3)

(ii) x2 – 2x = (–2) (3 – x)

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)

(iv) (x – 3)(2x +1) = x(x + 5)

(v) (2x – 1)(x – 3) = (x + 5)(x – 1)

(vi) x2 + 3x + 1 = (x – 2)2

(vii) (x + 2)3 = 2x (x2 – 1)

(viii) x3 – 4x2 – x + 1 = (x – 2)3

Solutions:

(i) Given,

(x + 1)2 = 2(x – 3)

By using the formula for (a+b)= a2+2ab+b2

⇒ x2 + 2x + 1 = 2x – 6

⇒ x2 + 7 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

(ii) Given, x2 – 2x = (–2) (3 – x)

⇒ x 2x = -6 + 2x

⇒ x– 4x + 6 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

(iii) Given, (x – 2)(x + 1) = (x – 1)(x + 3)

By multiplication

⇒ x– x – 2 = x+ 2x – 3

⇒ 3x – 1 = 0

Since the above equation is not in the form of ax2 + bx + c = 0.

Therefore, the given equation is not a quadratic equation.

(iv) Given, (x – 3)(2x +1) = x(x + 5)

By multiplication

⇒ 2x– 5x – 3 = x+ 5x

⇒  x– 10x – 3 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is a quadratic equation.

(v) Given, (2x – 1)(x – 3) = (x + 5)(x – 1)

By multiplication

⇒ 2x– 7x + 3 = x+ 4x – 5

⇒ x– 11x + 8 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

(vi) Given, x2 + 3x + 1 = (x – 2)2

By using the formula for (a-b)2=a2-2ab+b2

⇒ x2 + 3x + 1 = x2 + 4 – 4x

⇒ 7x – 3 = 0

Since the above equation is not in the form of ax2 + bx + c = 0.

Therefore, the given equation is not a quadratic equation.

(vii) Given, (x + 2)3 = 2x(x2 – 1)

By using the formula for (a+b)= a3+b3+3ab(a+b)

⇒ x3 + 8 + x2 + 12x = 2x3 – 2x

⇒ x3 + 14x – 6x2 – 8 = 0

Since the above equation is not in the form of ax2 + bx + c = 0.

Therefore, the given equation is not a quadratic equation.

(viii) Given, x3 – 4x2 – x + 1 = (x – 2)3

By using the formula for (a-b)= a3-b3-3ab(a-b)

⇒  x3 – 4x2 – x + 1 = x3 – 8 – 6x + 12x

⇒ 2x2 – 13x + 9 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is a quadratic equation.

NCERT Class 10 Maths Textbook Chapter 4 With Answer Book PDF Free Download

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