Write the discriminant of the following quadratic equations:
(i)2x² − 5x + 3 = 0
(ii)x² + 2x + 4 = 0
(iii)(x − 1) (2x − 1) = 0
(iv x² − 2x + k = 0, k ∈ R
(v)√3x²+2√2x-2√3=0
(vi)x² − x + 1 = 0
Answers
SOLUTION :
Given : 2x² – 5x + 3 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 2 , b = -5 and c = 3
D(discriminant) = b² – 4ac
D = (-5)² – 4 x 2 x 3
D = 25 – 24 = 1
D = 1
Hence,the discriminant of the quadratic equation 2x² – 5x + 3 = 0 is 1.
2) Given : x² + 2x + 4 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 1 , b = 2 and c = 4
D(discriminant) = b² – 4ac
D = (2)² – 4 x 1 x 4
D = 4 – 16 = – 12
D = - 12
Hence,the discriminant of the quadratic equation x² + 2x + 4 = 0 is = – 12.
3) Given : (x -1) (2x -1) = 0
(x -1) (2x -1) = 0
2x² - x - 2x + 1 = 0
2x² - 3x + 1 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 2 , b = -3 , c = 1
D(discriminant) = b² – 4ac
D = (-3)² – 4 x 2 x 1
D = 9 – 8 = 1
D = 1
Hence,the discriminant of the quadratic equation (x -1) (2x -1) = 0 is = 1.
4) Given : x² – 2x + k = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 1 , b = -2 , and c = k
D(discriminant) = b² – 4ac
D = (-2)² – 4(1)(k)
D= 4 – 4k
Hence,the discriminant of the quadratic equation x² – 2x + k = 0 is (4 – 4k)
5) Given : 3√x² + 2√2x–2√3 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = √3 ,b = 2√2 and c = −2√3
D(discriminant) = b² – 4ac
D = (2√2)² –(4× √3 × -2√3)
D = 8 + 24 = 32
D = 32
Hence,the discriminant of the quadratic equation 3√x² + 2√2x–2√3 = 0 is 32.
6) Given : x² – x + 1 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a =1 , b = -1 and c = 1
D(discriminant) = b² – 4ac
D = (-1)² – 4 x 1 x 1
D = 1 – 4 = – 3
D = - 3
Hence,the discriminant of the quadratic equation x² – x + 1 = 0 is - 3.
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On comparing the given equation with ax² + bx + c = 0
Here, a = 2 , b = -5 and c = 3
D(discriminant) = b² – 4ac
D = (-5)² – 4 x 2 x 3
D = 25 – 24 = 1
D = 1
Hence the discriminant of the quadratic equation 2x² – 5x + 3 = 0 is 1.
2) Given : x² + 2x + 4 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 1 , b = 2 and c = 4
D(discriminant) = b² – 4ac
D = (2)² – 4 x 1 x 4
D = 4 – 16 = – 12
D = -12
Hence,the discriminant of the quadratic equation x² + 2x + 4 = 0 is = – 12.
3) Given : (x -1) (2x -1) = 0
(x -1) (2x -1) = 0
2x² - x - 2x + 1 = 0
2x² - 3x + 1 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 2 , b = -3 , c = 1
D(discriminant) = b² – 4ac
D = (-3)² – 4 x 2 x 1
D = 9 – 8 = 1
D = 1
Hence,the discriminant of the quadratic equation (x -1) (2x -1) = 0 is = 1
4) Given : x² – 2x + k = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 1 , b = -2 , and c = k
D(discriminant) = b² – 4ac
D = (-2)² – 4(1)(k)
D= 4 – 4k
Hence the discriminant of the quadratic equation x² – 2x + k = 0 is (4 – 4k)
5) Given : 3√x² + 2√2x–2√3 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = √3 ,b = 2√2 and c = −2√3
D(discriminant) = b² – 4ac
D = (2√2)² –(4× √3 × -2√3)
D = 8 + 24 = 32
D = 32
Hence the discriminant of the quadratic equation 3√x² + 2√2x–2√3 = 0 is 32.
6) Given : x² – x + 1 = 0
On comparing the given equation with ax² + bx + c = 0
Here, a =1 , b = -1 and c = 1
D(discriminant) = b² – 4ac
D = (-1)² – 4 x 1 x 1
D = 1 – 4 = – 3
D = - 3
Hence the discriminant of the quadratic equation x² – x + 1 = 0 is - 3.