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NCERT Class 10 Maths Textbook Chapter 4 With Answer Book PDF Free Download
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)
| Author | NCERT |
| Language | English |
| No. of Pages | 23 |
| PDF Size | 1783 KB |
| Category | Mathematics |
| Source/ Credits | ncert.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)2 = 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)
⇒ x2 – 2x = -6 + 2x
⇒ x2 – 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
⇒ x2 – x – 2 = x2 + 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
⇒ 2x2 – 5x – 3 = x2 + 5x
⇒ x2 – 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
⇒ 2x2 – 7x + 3 = x2 + 4x – 5
⇒ x2 – 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)3 = 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)3 = a3-b3-3ab(a-b)
⇒ x3 – 4x2 – x + 1 = x3 – 8 – 6x2 + 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