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To find the value of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( kx^2 + 1 = kx + 3x - 11x^2 \)[/tex] has real and equal roots, we need to transform and analyze the equation step-by-step. Let's proceed with solving this problem:
### Step 1: Rewrite the equation in standard quadratic form
First, let's rewrite the given equation into the standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
Given:
[tex]\[ kx^2 + 1 = kx + 3x - 11x^2 \][/tex]
Combine like terms and move all terms to one side of the equation:
[tex]\[ kx^2 + 1 - kx - 3x + 11x^2 = 0 \][/tex]
[tex]\[ (k + 11)x^2 - (k + 3)x + 1 = 0 \][/tex]
So the quadratic equation in standard form is:
[tex]\[ (k + 11)x^2 - (k + 3)x + 1 = 0 \][/tex]
### Step 2: Determine the condition for real and equal roots
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have real and equal roots, the discriminant must be zero. The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the quadratic equation [tex]\((k + 11)x^2 - (k + 3)x + 1 = 0\)[/tex], we identify the coefficients as follows:
[tex]\[ a = k + 11 \][/tex]
[tex]\[ b = -(k + 3) \][/tex]
[tex]\[ c = 1 \][/tex]
### Step 3: Set up the discriminant equation
Set the discriminant [tex]\(\Delta\)[/tex] to zero:
[tex]\[ \Delta = b^2 - 4ac = 0 \][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant:
[tex]\[ (-(k + 3))^2 - 4(k + 11) \cdot 1 = 0 \][/tex]
[tex]\[ (k + 3)^2 - 4(k + 11) = 0 \][/tex]
### Step 4: Solve for [tex]\( k \)[/tex]
Expand and simplify the equation:
[tex]\[ (k + 3)^2 - 4(k + 11) = 0 \][/tex]
[tex]\[ k^2 + 6k + 9 - 4k - 44 = 0 \][/tex]
[tex]\[ k^2 + 2k - 35 = 0 \][/tex]
This is a quadratic equation in terms of [tex]\( k \)[/tex]:
[tex]\[ k^2 + 2k - 35 = 0 \][/tex]
### Step 5: Factor the quadratic equation
Factorize the quadratic equation:
[tex]\[ (k + 7)(k - 5) = 0 \][/tex]
So, the solutions are:
[tex]\[ k + 7 = 0 \quad \text{or} \quad k - 5 = 0 \][/tex]
[tex]\[ k = -7 \quad \text{or} \quad k = 5 \][/tex]
### Conclusion
The values of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( kx^2 + 1 = kx + 3x - 11x^2 \)[/tex] has real and equal roots are [tex]\( -7 \)[/tex] and [tex]\( 5 \)[/tex].
So, the correct answer is:
(c) [tex]\( 5, -7 \)[/tex]
### Step 1: Rewrite the equation in standard quadratic form
First, let's rewrite the given equation into the standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
Given:
[tex]\[ kx^2 + 1 = kx + 3x - 11x^2 \][/tex]
Combine like terms and move all terms to one side of the equation:
[tex]\[ kx^2 + 1 - kx - 3x + 11x^2 = 0 \][/tex]
[tex]\[ (k + 11)x^2 - (k + 3)x + 1 = 0 \][/tex]
So the quadratic equation in standard form is:
[tex]\[ (k + 11)x^2 - (k + 3)x + 1 = 0 \][/tex]
### Step 2: Determine the condition for real and equal roots
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have real and equal roots, the discriminant must be zero. The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the quadratic equation [tex]\((k + 11)x^2 - (k + 3)x + 1 = 0\)[/tex], we identify the coefficients as follows:
[tex]\[ a = k + 11 \][/tex]
[tex]\[ b = -(k + 3) \][/tex]
[tex]\[ c = 1 \][/tex]
### Step 3: Set up the discriminant equation
Set the discriminant [tex]\(\Delta\)[/tex] to zero:
[tex]\[ \Delta = b^2 - 4ac = 0 \][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant:
[tex]\[ (-(k + 3))^2 - 4(k + 11) \cdot 1 = 0 \][/tex]
[tex]\[ (k + 3)^2 - 4(k + 11) = 0 \][/tex]
### Step 4: Solve for [tex]\( k \)[/tex]
Expand and simplify the equation:
[tex]\[ (k + 3)^2 - 4(k + 11) = 0 \][/tex]
[tex]\[ k^2 + 6k + 9 - 4k - 44 = 0 \][/tex]
[tex]\[ k^2 + 2k - 35 = 0 \][/tex]
This is a quadratic equation in terms of [tex]\( k \)[/tex]:
[tex]\[ k^2 + 2k - 35 = 0 \][/tex]
### Step 5: Factor the quadratic equation
Factorize the quadratic equation:
[tex]\[ (k + 7)(k - 5) = 0 \][/tex]
So, the solutions are:
[tex]\[ k + 7 = 0 \quad \text{or} \quad k - 5 = 0 \][/tex]
[tex]\[ k = -7 \quad \text{or} \quad k = 5 \][/tex]
### Conclusion
The values of [tex]\( k \)[/tex] for which the quadratic equation [tex]\( kx^2 + 1 = kx + 3x - 11x^2 \)[/tex] has real and equal roots are [tex]\( -7 \)[/tex] and [tex]\( 5 \)[/tex].
So, the correct answer is:
(c) [tex]\( 5, -7 \)[/tex]
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