1. Which of the following is not quadratic equation ????
(a) 2x² + 1 = 5x - 2x²
(b) (2x + 3)² = 2(x² + 5)
(c) (√2x + 3)² = 2x² - 9
(d) (x - 3)² = 5x² - 4x - 3
Answers
Answer:
Question :-
a) 2x² + 1 = 5x - 2x²
b) (2x + 3)² = 2(x² + 5)
c) (√2x + 3)² = 2x² - 9
d) (x - 3)² = 5x² - 4x - 3
Solution :-
a) 2x² + 1 = 5x - 2x²
➻ 2x² + 2x² + 1 = 5x
➻ 2x² + 2x² - 5x + 1 = 0
➻ 4x² - 5x + 1 = 0
b) (2x + 3)² = 2(x² + 5)
Formula used :-
➪ (a + b)²
➭ a² + 2ab + b²
➻ (2x)² + 2 × 2x × 3 + (3)² = 2x² + 10
➻ 4x² + 12x + 9 = 2x² + 10
➻ 4x² - 2x² + 12x + 9 - 10 = 0
➻ 2x² + 12x - 1 = 0
c) (√2x + 3)² = 2x² - 9
Formula used :-
➪ (a + b)²
➭ a² + 2ab + b²
➻ (√2x)² + 2 × √2x × 3 + (3)² = 2x² - 9
➻ 2x² + 6√2x + 9 = 2x² - 9
➻ 2x² - 2x² + 6√2x + 9 + 9 = 0
➻ 6√2x + 18 = 0
➻ 6(√2x + 3) = 0
➻ √2x + 3 = 0
d) (x - 3)² = 5x² - 4x - 3
Formula used :-
➪ (a - b)²
➭ a² - 2ab + b²
➻ (x)² - 2 × x × 3 + (3)² = 5x² - 4x - 3
➻ x² - 6x + 9 = 5x² - 4x - 3
➻ 5x² - x² - 4x + 6x - 3 - 9 = 0
➻ 4x² + 2x - 12 = 0
➻ 2(2x² + x - 6) = 0
➻ 2x² + x - 6 = 0
Now from the simplified from of the given equations it is evident that (c) is not a quadratic equation.
Solution!!
We have to find out which of the following is not a quadratic equation. We know that the quadratic equation is in the form of ax² + bx + c = 0. So, whichever equation will not be suitable in this form will not be a quadratic equation.
(a) 2x² + 1 = 5x - 2x²
2x² + 1 + 2x² = 5x
4x² + 1 = 5x
4x² + 1 - 5x = 0
4x² - 5x + 1 = 0
As we can observe that this is similar to the form of the quadratic equation where a is 4, b is -5 and c is 1. Hence, this is a quadratic equation.
(b) (2x + 3)² = 2(x² + 5)
(2x)² + (3)² + 2(2x)(3) = 2x² + 10
4x² + 9 + 12x = 2x² + 10
4x² + 9 + 12x - 2x² = 10
2x² + 9 + 12x = 10
2x² + 9 + 12x - 10 = 0
2x² + 12x - 1 = 0
This too, is similar to the form of the quadratic equation where a is 2, b is 12 and c is -1. Hence, this too, is a quadratic equation.
(c) (√2x + 3)² = 2x² - 9
(√2x)² + (3)² + 2(√2x)(3) = 2x² - 9
2x² + 9 + 6√2x = 2x² - 9
2x² + 9 + 6√2x - 2x² = -9
9 + 6√2x + 9 = 0
6√2x + 18 = 0
This is not similar to the form of the quadratic equation. This is in the form of bx + c = 0 whereas the quadratic equation is in the form of ax² + bx + c = 0. Hence, this is not a quadratic equation.
(d) (x - 3)² = 5x² - 4x - 3
(x)² + (3)² - 2(x)(3) = 5x² - 4x - 3
x² + 9 - 6x = 5x² - 4x - 3
9 - 6x = - x² + 5x² - 4x - 3
9 - 6x = 4x² - 4x - 3
- 6x = 4x² - 4x - 3 - 9
- 6x = 4x² - 4x - 12
0 = 4x² - 4x - 12 + 6x
0 = 4x² + 2x - 12
4x² + 2x - 12 = 0
This is also in the form of quadratic equation where a is 4, b is 2 and c is -12. Hence, this is a quadratic equation.
Identities used:
→ (a + b)² = a² + b² + 2ab
→ (a - b)² = a² + b² - 2ab