Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. I didn't really understand what that meant.
x^2-23x+c
x^2-(2/7)x+c
take half the middle term coefficient, square it
x^2 - 23x + 526/4
= (x- 23/2)^2
looks like you learning the process of "completing the square"
here is an easier one ...
x^2 + 10x + c
or
x^2 + 10x + 25 ,,,,,(half of 10 is 5, 5^2 = 25)
= (x+5)^2
expand (x+5)^2 to see why this happens
Now try your own second question and let me know what you got for c .
To find the value of c that makes the expression a perfect square trinomial, you can use the following steps:
1. Identify the coefficient of the x term. In the first expression, the coefficient is -23x, and in the second expression, the coefficient is -(2/7)x.
2. Take half of the coefficient (divide it by 2). In the first expression, (-23x)/2 = -11.5x, and in the second expression, (-(2/7)x)/2 = -(1/7)x.
3. Square the result from step 2. In the first expression, (-11.5x)^2 = 132.25x^2, and in the second expression, (-(1/7)x)^2 = (1/49)x^2.
4. The squared result from step 3 represents the missing term in the trinomial to make it a perfect square trinomial. So, in the first expression, c = 132.25, and in the second expression, c = (1/49).
To write the expression as the square of a binomial, you'll follow these steps:
1. Start with the given expression: x^2 - 23x + c (or x^2 - (2/7)x + c).
2. Take the square root of the constant term c found in the previous step. In the first expression, √132.25 = 11.5, and in the second expression, √(1/49) = 1/7.
3. Write the binomial as (x - square root of c)^2. Substitute the value found from step 2. In the first expression, (x - 11.5)^2, and in the second expression, (x - 1/7)^2.
Therefore, the expression x^2 - 23x + c can be written as (x - 11.5)^2, and the expression x^2 - (2/7)x + c can be written as (x - 1/7)^2.
To find the value of c that makes the expression a perfect square trinomial, we need to ensure that the coefficient of the linear term (x term) is twice the product of the square root of the coefficient of the quadratic term (x^2 term) and the square root of c.
Let's break down the steps:
1. For the first expression, x^2 - 23x + c:
- The coefficient of the quadratic term is 1 (x^2 term).
- The coefficient of the linear term is -23 (x term).
- The square root of the coefficient of the quadratic term is 1.
- Let's assume that the square root of c is d.
2. According to the formula, the coefficient of the linear term (-23) should be twice the product of the square root of the quadratic term (1) and the square root of c (d).
- So, we have: -23 = 2(1)(d)
- Simplifying the equation: -23 = 2d
- Divide both sides by 2: d = -23/2
Therefore, the value of c that makes the expression a perfect square trinomial in the first expression is c = (-23/2)^2 = 529/4.
To write the expression as the square of a binomial, we can use the value of c:
x^2 - 23x + 529/4 can be written as (x - 23/2)^2.
For the second expression, x^2 - (2/7)x + c, the process is similar and the value of c would be c = (2/14)^2 = 1/49.
Thus, the second expression can be written as (x - 1/7)^2.