Which the following is a quadratic equation
(a) x² + 2x + 1 = (4 -x)² + 3
(b) -2x² = (5 - x) (2x - 2/5)
(c) (k + 1) x² + 3/2 x = 7, where k is not equal to 1
d) x³ - x² = (x-1) ³
Answers
Step-by-step explanation:
a)
x^2+2x+1=16-8x+x^2+3
x^2+2x+1=19-8x+x^2
10x=20
x=2
it is not a quadratic equation
b)
-2x^2=(5-x)(10x-2/5)
-2x^2=50x-2x^2+2x/5
50x+2x/5=0
it is not a quadratic equation
c)
9/4-4(k+1)(-7)=0
9/4-28k-28=0
it is not a quadratic equation
d)
x^3-x^2=x^3-1-3x(x-1)
x^3-x^2=x^3-1-3x^2+3x
2x^2-3x+1=0
it is a quadratic eauation
*Correct Question :-
Which the following is a quadratic equation
(a) x² + 2x + 1 = (4 -x)² + 3
(b) -2x² = (5 - x) (2x - 2/5)
(c) (k + 1) x² + 3/2 x = 7, where k is not equal to -1*
d) x³ - x² = (x-1) ³
AnswEr :-
- option c and d
Given :-
- (a) x² + 2x + 1 = (4 -x)² + 3
- (b) -2x² = (5 - x) (2x - 2/5)
- (c) (k + 1) x² + 3/2 x = 7, where k is not equal to (-1)*
- d) x³ - x² = (x-1) ³
To Find :-
- The quadratic equation
Solution :-
➤ We know, the common formation of a quadratic equation as,
ax² + bx + c = 0 (a ≠ 0 ; a,b,c are real numbers)
➤ So now, we will check all the options given in the question if there is any equation which matches with quadratic equation formation.
➤ Let's do it :-
(a) x² + 2x + 1 = (4 - x)² + 3
➻ x² + 2x + 1 = 4² - 2 × 4x + x² + 3 [using (a-b)² formula)
➻ x² + 2x + 1 - 16 + 8x - x² - 3 = 0
➻ 10x - 18 = 0 {not a quadratic equation}
━━━━━━━━━━━━━━━━━━━━━━━━━━⠀
(b) -2x² = (5 - x) (2x - 2/5)
➞ -2x² = 10x - 10 - 2x² + (2x/5)
➞ -2x² - 10x + 10 + 2x² - (2x/5) = 0
➞ - (10x + 2x/5) + 10 = 0
➞ - 52x/5 + 10 = 0 {not a quadratic equation}
━━━━━━━━━━━━━━━━━━━━━━━━━━⠀
(c) (k + 1) x² + 3/2 x = 7, where k is not equal to -1
➙ (k + 1) x² + 3/2 x - 7 = 0 {formation of a quadratic equation}
* as k ≠ -1 ; therefore, (k + 1) ≠ 0
━━━━━━━━━━━━━━━━━━━━━━━━━━⠀
(d) x³ - x² = (x-1) ³
➠ x³ - x² = x³ - 1³ + 3x ( x - 1 )
➠ x³ - x² - x³ + 1³ - 3x² + 3x = 0
➠ - 4x² + 3x +1 = 0
➠ 4x² - 3x - 1 = 0 { a quadratic equation}
━━━━━━━━━━━━━━━━━━━━━━━━━━⠀
Thus, we are done with the calculation!