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If the equation kx² – 6x + 2 = 0 has equal roots, then the value of k is:
For equal roots, discriminant must be zero: b² - 4ac = 0 Given equation: kx² - 6x + 2 = 0 Here: a = k b = -6 c = 2 Putting in discriminant: (-6)² - 4(k)(2) = 0 36 - 8k = 0 8k = 36 k = 9 To check: When k = 9: 9x² - 6x + 2 = 0 Using quadratic formula: x = [6 ± √(36 - 72)]/18 x = [6 ± 0]/18 x = 1/3 (reRead more
For equal roots, discriminant must be zero:
b² – 4ac = 0
Given equation: kx² – 6x + 2 = 0
Here:
a = k
b = -6
c = 2
Putting in discriminant:
(-6)² – 4(k)(2) = 0
36 – 8k = 0
8k = 36
k = 9
To check:
When k = 9:
9x² – 6x + 2 = 0
Using quadratic formula:
x = [6 ± √(36 – 72)]/18
x = [6 ± 0]/18
x = 1/3 (repeated root)
Therefore, 9 is the answer.
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The quadratic equation whose roots are 5 and -2 is:
The quadratic equation whose roots are 5 and -2 is: x² - 3x - 10 = 0 Let's verify: If α = 5 and β = -2 are roots then: Sum of roots = -(coefficient of x)/coefficient of x² α + β = -b/a = 3 Product of roots = constant term/coefficient of x² α × β = c/a = -10 Therefore x² - 3x - 10 = 0 is correct as:Read more
The quadratic equation whose roots are 5 and -2 is: x² – 3x – 10 = 0
Let’s verify:
If α = 5 and β = -2 are roots then:
Sum of roots = -(coefficient of x)/coefficient of x²
α + β = -b/a = 3
Product of roots = constant term/coefficient of x²
α × β = c/a = -10
Therefore x² – 3x – 10 = 0 is correct as:
– coefficient of x: -(α + β) = -3
– constant term: α × β = -10
Hence option x² – 3x – 10 = 0 is correct.
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If one root of the equation x² + px + 12 = 0 is -3, then the value of p is:
Given equation: x² + px + 12 = 0 One root is -3 Since -3 is a root it must satisfy the equation: (-3)² + p(-3) + 12 = 0 Simplifying: 9 - 3p + 12 = 0 21 - 3p = 0 -3p = -21 p = 7 To verify: When p = 7: x² + 7x + 12 = 0 Roots are -3 and -4 One root is indeed -3 Hence, 7 is the correct answer. Click herRead more
Given equation: x² + px + 12 = 0
One root is -3
Since -3 is a root it must satisfy the equation:
(-3)² + p(-3) + 12 = 0
Simplifying:
9 – 3p + 12 = 0
21 – 3p = 0
-3p = -21
p = 7
To verify:
When p = 7:
x² + 7x + 12 = 0
Roots are -3 and -4
One root is indeed -3
Hence, 7 is the correct answer.
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The roots of the equation 3x² – 2x – 1 = 0 using the quadratic formula are:
Given equation: 3x² - 2x - 1 = 0 Using quadratic formula: x = [-b ± √(b² - 4ac)]/2a Here: a = 3 b = -2 c = -1 Substituting: x = [2 ± √(4 - 4(3)(-1))]/6 x = [2 ± √(4 + 12)]/6 x = [2 ± √16]/6 x = [2 ± 4]/6 For + sign: x = (2 + 4)/6 x = 6/6 x = 1 For - sign: x = (2 - 4)/6 x = -2/6 x = -1/3 Therefore roRead more
Given equation: 3x² – 2x – 1 = 0
Using quadratic formula:
x = [-b ± √(b² – 4ac)]/2a
Here:
a = 3
b = -2
c = -1
Substituting:
x = [2 ± √(4 – 4(3)(-1))]/6
x = [2 ± √(4 + 12)]/6
x = [2 ± √16]/6
x = [2 ± 4]/6
For + sign:
x = (2 + 4)/6
x = 6/6
x = 1
For – sign:
x = (2 – 4)/6
x = -2/6
x = -1/3
Therefore roots are: 1 and -1/3
To verify:
3(1)² – 2(1) – 1 = 0
3(-1/3)² – 2(-1/3) – 1 = 0
Hence, 1, -1/3 are the correct roots.
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The nature of the roots of the equation x² – 4x + 5 = 0 is:
Given equation: x² - 4x + 5 = 0 For nature of roots check discriminant: b² - 4ac Here: a = 1 b = -4 c = 5 Discriminant = (-4)² - 4(1)(5) = 16 - 20 = -4 Since discriminant < 0: The roots are imaginary (or complex conjugates) We can verify: Using quadratic formula: x = [4 ± √(-4)]/2 x = 2 ± i ThereRead more
Given equation: x² – 4x + 5 = 0
For nature of roots check discriminant:
b² – 4ac
Here:
a = 1
b = -4
c = 5
Discriminant = (-4)² – 4(1)(5)
= 16 – 20
= -4
Since discriminant < 0:
The roots are imaginary (or complex conjugates)
We can verify:
Using quadratic formula:
x = [4 ± √(-4)]/2
x = 2 ± i
Therefore roots are complex conjugates: 2 + i and 2 – i
Hence, the nature of roots is Imaginary.
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