Find the value of k so that the equation x^2+kx+k-1 = 0 has one real root.
person
brightness_auto
7 Answers
For an equation to have one real root, the discriminant (the value inside the square root in the quadratic formula) must be equal to zero. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
In your equation, x^2 + kx + (k - 1) = 0, the coefficients are:
a = 1
b = k
c = k - 1
Now, let's calculate the discriminant:
Discriminant = b^2 - 4ac = k^2 - 4(1)(k - 1) = k^2 - 4k + 4
To have one real root, the discriminant must be equal to zero:
k^2 - 4k + 4 = 0
Now, solve this quadratic equation for k:
(k - 2)(k - 2) = 0
(k - 2)^2 = 0
Taking the square root of both sides:
k - 2 = 0
k = 2
So, the value of k that makes the equation x^2 + kx + k - 1 = 0 have one real root is k = 2.
x = (-b ± √(b^2 - 4ac)) / (2a)
In your equation, x^2 + kx + (k - 1) = 0, the coefficients are:
a = 1
b = k
c = k - 1
Now, let's calculate the discriminant:
Discriminant = b^2 - 4ac = k^2 - 4(1)(k - 1) = k^2 - 4k + 4
To have one real root, the discriminant must be equal to zero:
k^2 - 4k + 4 = 0
Now, solve this quadratic equation for k:
(k - 2)(k - 2) = 0
(k - 2)^2 = 0
Taking the square root of both sides:
k - 2 = 0
k = 2
So, the value of k that makes the equation x^2 + kx + k - 1 = 0 have one real root is k = 2.
For the equation to have one real root, the discriminant must be equal to zero. The discriminant is given by the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
In this case, a = 1, b = k, and c = k-1. So, the discriminant becomes:
D = k^2 - 4(1)(k-1)
D = k^2 - 4k + 4
For the equation to have one real root, the discriminant must be equal to zero:
D = 0
k^2 - 4k + 4 = 0
We can solve this quadratic equation by factoring:
(k - 2)(k - 2) = 0
(k - 2)^2 = 0
Taking the square root of both sides, we get:
k - 2 = 0
k = 2
Therefore, the value of k that makes the equation have one real root is k = 2.
In this case, a = 1, b = k, and c = k-1. So, the discriminant becomes:
D = k^2 - 4(1)(k-1)
D = k^2 - 4k + 4
For the equation to have one real root, the discriminant must be equal to zero:
D = 0
k^2 - 4k + 4 = 0
We can solve this quadratic equation by factoring:
(k - 2)(k - 2) = 0
(k - 2)^2 = 0
Taking the square root of both sides, we get:
k - 2 = 0
k = 2
Therefore, the value of k that makes the equation have one real root is k = 2.
For the equation to have one real root, the discriminant (b^2 - 4ac) must be zero. In this case, a = 1, b = k, and c = k - 1. So, we have k^2 - 4(1)(k - 1) = 0. Simplifying, we get k^2 - 4k + 4 = 0. Factoring, (k - 2)^2 = 0. Therefore, k = 2.
To find the value of k such that the equation 2kx - k = 0 has one real root, we can use the discriminant of the quadratic equation.
The discriminant (D) is given by b^2 - 4ac, where the quadratic equation is in the form ax^2 + bx + c = 0.
Comparing the given equation 2kx - k = 0 with the quadratic equation ax^2 + bx + c = 0, we have:
a = 2k
b = 0
c = -k
Substituting these values into the discriminant formula, we get:
D = b^2 - 4ac
= 0^2 - 4(2k)(-k)
= 0 + 8k^2
= 8k^2
For the equation to have one real root, the discriminant D must be equal to zero. Therefore, we have:
8k^2 = 0
Solving this equation, we find that k = 0. So, the value of k that satisfies the condition is k = 0.
The discriminant (D) is given by b^2 - 4ac, where the quadratic equation is in the form ax^2 + bx + c = 0.
Comparing the given equation 2kx - k = 0 with the quadratic equation ax^2 + bx + c = 0, we have:
a = 2k
b = 0
c = -k
Substituting these values into the discriminant formula, we get:
D = b^2 - 4ac
= 0^2 - 4(2k)(-k)
= 0 + 8k^2
= 8k^2
For the equation to have one real root, the discriminant D must be equal to zero. Therefore, we have:
8k^2 = 0
Solving this equation, we find that k = 0. So, the value of k that satisfies the condition is k = 0.
To have a quadratic condition with one genuine root, the discriminant (the worth inside the square base of the quadratic equation) should be equivalent to nothing. The discriminant is given by: Discriminant (D) = b^2 - 4ac In your situation x^2 + kx + (k - 1) = 0, you have: a = 1 b = k c = k - 1 Presently, set the discriminant equivalent to nothing and address for k: D = k^2 - 4(1)(k - 1) = 0 Streamline and tackle: k^2 - 4k + 4 = 0 Presently, we have a quadratic condition in k. To find the worth of k, you can factor the condition: (k - 2)(k - 2) = 0 Since (k - 2) is squared, it shows up two times, demonstrating that there is one genuine root with a variety of 2 (a rehashed root). Thus, the condition x^2 + kx + (k - 1) = 0 has one genuine root when k = 2.
For the quadratic equation
�
2
+
�
�
+
�
−
1
=
0
x
2
+kx+k−1=0 to have one real root, the discriminant (
�
2
−
4
�
�
b
2
−4ac) should be equal to zero. In this case,
�
=
1
a=1,
�
=
�
b=k, and
�
=
�
−
1
c=k−1. So:
�
2
−
4
�
�
=
�
2
−
4
(
1
)
(
�
−
1
)
b
2
−4ac=k
2
−4(1)(k−1)
Set
�
2
−
4
�
+
4
=
0
k
2
−4k+4=0 to find the value of
�
k. Factoring, you get
(
�
−
2
)
2
=
0
(k−2)
2
=0. Therefore,
�
=
2
k=2 is the solution.
�
2
+
�
�
+
�
−
1
=
0
x
2
+kx+k−1=0 to have one real root, the discriminant (
�
2
−
4
�
�
b
2
−4ac) should be equal to zero. In this case,
�
=
1
a=1,
�
=
�
b=k, and
�
=
�
−
1
c=k−1. So:
�
2
−
4
�
�
=
�
2
−
4
(
1
)
(
�
−
1
)
b
2
−4ac=k
2
−4(1)(k−1)
Set
�
2
−
4
�
+
4
=
0
k
2
−4k+4=0 to find the value of
�
k. Factoring, you get
(
�
−
2
)
2
=
0
(k−2)
2
=0. Therefore,
�
=
2
k=2 is the solution.
To find the value of \( k \) such that the equation \( x^2 + kx + k - 1 = 0 \) has one real root, we can use the discriminant of the quadratic equation.
The discriminant, denoted by \( \Delta \), is given by the formula:
\[ \Delta = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).
In our equation, \( a = 1 \), \( b = k \), and \( c = k - 1 \).
So, substituting these values into the discriminant formula:
\[ \Delta = k^2 - 4(1)(k - 1) \]
\[ \Delta = k^2 - 4k + 4 \]
For the equation to have one real root, the discriminant must be equal to zero. Therefore:
\[ k^2 - 4k + 4 = 0 \]
This is a quadratic equation in \( k \). We can solve it by factoring or using the quadratic formula. Since this equation factors nicely, let's factor it:
\[ (k - 2)^2 = 0 \]
Now, take the square root of both sides:
\[ k - 2 = 0 \]
\[ k = 2 \]
So, the value of \( k \) that makes the equation \( x^2 + kx + k - 1 = 0 \) have one real root is \( k = 2 \).
The discriminant, denoted by \( \Delta \), is given by the formula:
\[ \Delta = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).
In our equation, \( a = 1 \), \( b = k \), and \( c = k - 1 \).
So, substituting these values into the discriminant formula:
\[ \Delta = k^2 - 4(1)(k - 1) \]
\[ \Delta = k^2 - 4k + 4 \]
For the equation to have one real root, the discriminant must be equal to zero. Therefore:
\[ k^2 - 4k + 4 = 0 \]
This is a quadratic equation in \( k \). We can solve it by factoring or using the quadratic formula. Since this equation factors nicely, let's factor it:
\[ (k - 2)^2 = 0 \]
Now, take the square root of both sides:
\[ k - 2 = 0 \]
\[ k = 2 \]
So, the value of \( k \) that makes the equation \( x^2 + kx + k - 1 = 0 \) have one real root is \( k = 2 \).