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### 17) For what values of [tex]\( m \)[/tex] would the equation [tex]\( x^2 + 12x + m = 0 \)[/tex] have one real root?
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have one real root, its discriminant must be zero. The discriminant for a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In the given equation [tex]\( x^2 + 12x + m = 0 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 12 \)[/tex]
- [tex]\( c = m \)[/tex]
Set the discriminant to zero and solve for [tex]\( m \)[/tex]:
[tex]\[ \Delta = 12^2 - 4 \cdot 1 \cdot m = 0 \][/tex]
[tex]\[ 144 - 4m = 0 \][/tex]
[tex]\[ 144 = 4m \][/tex]
[tex]\[ m = \frac{144}{4} \][/tex]
[tex]\[ m = 36 \][/tex]
So, the equation [tex]\( x^2 + 12x + m = 0 \)[/tex] will have one real root when [tex]\( m = 36 \)[/tex].
### 18) For what value(s) of [tex]\( k \)[/tex] would the equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex] have one real root?
Again, for the quadratic equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex] to have one real root, its discriminant must be zero. The discriminant for a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In the given equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex], we have:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = k \)[/tex]
- [tex]\( c = 3 \)[/tex]
Set the discriminant to zero and solve for [tex]\( k \)[/tex]:
[tex]\[ \Delta = k^2 - 4 \cdot 3 \cdot 3 = 0 \][/tex]
[tex]\[ k^2 - 36 = 0 \][/tex]
[tex]\[ k^2 = 36 \][/tex]
[tex]\[ k = \pm \sqrt{36} \][/tex]
[tex]\[ k = \pm 6 \][/tex]
So, the equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex] will have one real root when [tex]\( k = 6 \)[/tex] or [tex]\( k = -6 \)[/tex].
### Summary:
- For [tex]\( x^2 + 12x + m = 0 \)[/tex], the value of [tex]\( m \)[/tex] that makes the equation have one real root is [tex]\( m = 36 \)[/tex].
- For [tex]\( 3x^2 + kx + 3 = 0 \)[/tex], the values of [tex]\( k \)[/tex] that make the equation have one real root are [tex]\( k = 6 \)[/tex] and [tex]\( k = -6 \)[/tex].
### 17) For what values of [tex]\( m \)[/tex] would the equation [tex]\( x^2 + 12x + m = 0 \)[/tex] have one real root?
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have one real root, its discriminant must be zero. The discriminant for a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In the given equation [tex]\( x^2 + 12x + m = 0 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 12 \)[/tex]
- [tex]\( c = m \)[/tex]
Set the discriminant to zero and solve for [tex]\( m \)[/tex]:
[tex]\[ \Delta = 12^2 - 4 \cdot 1 \cdot m = 0 \][/tex]
[tex]\[ 144 - 4m = 0 \][/tex]
[tex]\[ 144 = 4m \][/tex]
[tex]\[ m = \frac{144}{4} \][/tex]
[tex]\[ m = 36 \][/tex]
So, the equation [tex]\( x^2 + 12x + m = 0 \)[/tex] will have one real root when [tex]\( m = 36 \)[/tex].
### 18) For what value(s) of [tex]\( k \)[/tex] would the equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex] have one real root?
Again, for the quadratic equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex] to have one real root, its discriminant must be zero. The discriminant for a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In the given equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex], we have:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = k \)[/tex]
- [tex]\( c = 3 \)[/tex]
Set the discriminant to zero and solve for [tex]\( k \)[/tex]:
[tex]\[ \Delta = k^2 - 4 \cdot 3 \cdot 3 = 0 \][/tex]
[tex]\[ k^2 - 36 = 0 \][/tex]
[tex]\[ k^2 = 36 \][/tex]
[tex]\[ k = \pm \sqrt{36} \][/tex]
[tex]\[ k = \pm 6 \][/tex]
So, the equation [tex]\( 3x^2 + kx + 3 = 0 \)[/tex] will have one real root when [tex]\( k = 6 \)[/tex] or [tex]\( k = -6 \)[/tex].
### Summary:
- For [tex]\( x^2 + 12x + m = 0 \)[/tex], the value of [tex]\( m \)[/tex] that makes the equation have one real root is [tex]\( m = 36 \)[/tex].
- For [tex]\( 3x^2 + kx + 3 = 0 \)[/tex], the values of [tex]\( k \)[/tex] that make the equation have one real root are [tex]\( k = 6 \)[/tex] and [tex]\( k = -6 \)[/tex].
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