If 3x2-2x+k is a perfect square,what must be the value of k?
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To make the expression 3x^2 - 2x + k a perfect square, it should be in the form of (ax + b)^2.
Let's find what (ax + b)^2 expands to:
(ax + b)^2 = a^2x^2 + 2abx + b^2
Comparing this to the expression 3x^2 - 2x + k, we can see that:
a^2 = 3 (to match the x^2 coefficient)
2ab = -2 (to match the x coefficient)
b^2 = k (to match the constant term)
From the second equation, we can solve for "a" and "b":
2ab = -2
2ab = -2
ab = -1
Now, from the first equation:
a^2 = 3
a = √3
Now, we can calculate "b" using the fact that ab = -1:
√3 * b = -1
b = -1/√3
Now, we know "b^2" should equal "k":
b^2 = (-1/√3)^2 = 1/3
So, the value of k that makes 3x^2 - 2x + k a perfect square is k = 1/3.
Let's find what (ax + b)^2 expands to:
(ax + b)^2 = a^2x^2 + 2abx + b^2
Comparing this to the expression 3x^2 - 2x + k, we can see that:
a^2 = 3 (to match the x^2 coefficient)
2ab = -2 (to match the x coefficient)
b^2 = k (to match the constant term)
From the second equation, we can solve for "a" and "b":
2ab = -2
2ab = -2
ab = -1
Now, from the first equation:
a^2 = 3
a = √3
Now, we can calculate "b" using the fact that ab = -1:
√3 * b = -1
b = -1/√3
Now, we know "b^2" should equal "k":
b^2 = (-1/√3)^2 = 1/3
So, the value of k that makes 3x^2 - 2x + k a perfect square is k = 1/3.
To determine the value of k, we need to rewrite the expression as a perfect square.
We know that (x + a)^2 = x^2 + 2ax + a^2.
Comparing this to 3x^2 - 2x + k, we can see that the coefficient of x should be 2a, which means 2a = -2. Solving for a gives us a = -1.
Now, we can rewrite the expression:
3x^2 - 2x + k = (x - 1)^2.
Expanding (x - 1)^2, we get:
x^2 - 2x + 1.
Comparing this to 3x^2 - 2x + k, we can see that they are equal when k = 1.
Therefore, the value of k must be 1 in order for 3x^2 - 2x + k to be a perfect square.
We know that (x + a)^2 = x^2 + 2ax + a^2.
Comparing this to 3x^2 - 2x + k, we can see that the coefficient of x should be 2a, which means 2a = -2. Solving for a gives us a = -1.
Now, we can rewrite the expression:
3x^2 - 2x + k = (x - 1)^2.
Expanding (x - 1)^2, we get:
x^2 - 2x + 1.
Comparing this to 3x^2 - 2x + k, we can see that they are equal when k = 1.
Therefore, the value of k must be 1 in order for 3x^2 - 2x + k to be a perfect square.
To determine the value of k for which the expression 3x^2 - 2x + k is a perfect square, you need to complete the square. A perfect square expression can be written in the form (ax + b)^2.
Start by completing the square for the given expression:
3x^2 - 2x + k
First, factor out the coefficient of x^2, which is 3:
3(x^2 - (2/3)x) + k
Now, to complete the square, you'll need to add and subtract the square of half the coefficient of x, which is (-2/3)^2 = 4/9:
3(x^2 - (2/3)x + 4/9 - 4/9) + k
Now, rewrite the expression:
3[(x - 2/3)^2 - 4/9] + k
Expand the square:
3(x - 2/3)^2 - 4/3 + k
Now, for the expression to be a perfect square, the constant term (-4/3 + k) must be 0 because perfect squares have no constant term.
So, set -4/3 + k = 0 and solve for k:
k = 4/3
Therefore, the value of k for which the expression 3x^2 - 2x + k is a perfect square is k = 4/3.
Start by completing the square for the given expression:
3x^2 - 2x + k
First, factor out the coefficient of x^2, which is 3:
3(x^2 - (2/3)x) + k
Now, to complete the square, you'll need to add and subtract the square of half the coefficient of x, which is (-2/3)^2 = 4/9:
3(x^2 - (2/3)x + 4/9 - 4/9) + k
Now, rewrite the expression:
3[(x - 2/3)^2 - 4/9] + k
Expand the square:
3(x - 2/3)^2 - 4/3 + k
Now, for the expression to be a perfect square, the constant term (-4/3 + k) must be 0 because perfect squares have no constant term.
So, set -4/3 + k = 0 and solve for k:
k = 4/3
Therefore, the value of k for which the expression 3x^2 - 2x + k is a perfect square is k = 4/3.
If 3x^2 - 2x + k is a perfect square, the value of k can be determined by considering the properties of perfect squares. Specifically, the quadratic expression must be the square of a binomial.
To find the value of k, we can take the coefficient of the linear term (-2x) and divide it by 2, and then square the result. So, (-2/2)^2 = (-1)^2 = 1.
Therefore, the value of k must be 1 for 3x^2 - 2x + k to be a perfect square.
To find the value of k, we can take the coefficient of the linear term (-2x) and divide it by 2, and then square the result. So, (-2/2)^2 = (-1)^2 = 1.
Therefore, the value of k must be 1 for 3x^2 - 2x + k to be a perfect square.
First, recognize that a perfect square expression usually takes the form of (ax + b)^2, where a and b are constants. You need to find these constants a and b.
Start with the given expression: 3x^2 - 2x + k.
Take the square root of the coefficient of x^2 (which is 3). This gives you a = √3.
Next, you need to find b. To do this, notice that (ax + b)^2 = a^2x^2 + 2abx + b^2.
Compare the linear term (-2x) in the given expression to the linear term (2abx) in the perfect square form. This means that 2ab = -2.
Solve for b:
2ab = -2
2(√3)b = -2
b = -1/√3
Now that you have found the values of a and b, you can express the perfect square:
(ax + b)^2 = (√3x - 1/√3)^2 = 3x - 2/√3 + 1/3.
So, for the expression 3x^2 - 2x + k to be a perfect square, the value of k must be 1/3.
Start with the given expression: 3x^2 - 2x + k.
Take the square root of the coefficient of x^2 (which is 3). This gives you a = √3.
Next, you need to find b. To do this, notice that (ax + b)^2 = a^2x^2 + 2abx + b^2.
Compare the linear term (-2x) in the given expression to the linear term (2abx) in the perfect square form. This means that 2ab = -2.
Solve for b:
2ab = -2
2(√3)b = -2
b = -1/√3
Now that you have found the values of a and b, you can express the perfect square:
(ax + b)^2 = (√3x - 1/√3)^2 = 3x - 2/√3 + 1/3.
So, for the expression 3x^2 - 2x + k to be a perfect square, the value of k must be 1/3.
To determine the value of \( k \) for which the expression \( 3x^2 - 2x + k \) is a perfect square, we can consider the general form of a perfect square trinomial.
A perfect square trinomial can be expressed as \( (ax + b)^2 \), where \( a \) is the coefficient of \( x^2 \), and \( b \) is half the coefficient of \( x \).
In our case, the given expression is \( 3x^2 - 2x + k \). Comparing this with \( (ax + b)^2 \), we can identify that \( a = 3 \) and \( b = -1 \) (half of the coefficient of \( x \)).
Now, expand \( (ax + b)^2 \) to find the perfect square trinomial:
\[ (3x - 1)^2 = 9x^2 - 6x + 1 \]
For this to match the given expression \( 3x^2 - 2x + k \), we must have \( k = 1 \).
Therefore, the value of \( k \) for which \( 3x^2 - 2x + k \) is a perfect square is \( k = 1 \).
A perfect square trinomial can be expressed as \( (ax + b)^2 \), where \( a \) is the coefficient of \( x^2 \), and \( b \) is half the coefficient of \( x \).
In our case, the given expression is \( 3x^2 - 2x + k \). Comparing this with \( (ax + b)^2 \), we can identify that \( a = 3 \) and \( b = -1 \) (half of the coefficient of \( x \)).
Now, expand \( (ax + b)^2 \) to find the perfect square trinomial:
\[ (3x - 1)^2 = 9x^2 - 6x + 1 \]
For this to match the given expression \( 3x^2 - 2x + k \), we must have \( k = 1 \).
Therefore, the value of \( k \) for which \( 3x^2 - 2x + k \) is a perfect square is \( k = 1 \).