Austin and Tamara each completed the square of the quadratic equation 0=3x square - 15x + 6
Austin 0=3(x-5/2) square-51/4
Tamara 0=(x-5/2)square-17/4
Who is correct and show why
mine:
3x^2 - 15x + 6 = 0
x^2 - 5x + .... = -2 + .....
x^2 - 5x + 25/4 = -2 + 25/4
(x-5/2)^2 = 17/4
(x - 5/2)^2 - 17/4 = 0
suppose I multiply by 3
3(x-5/2)^2 - 51/4 = 0
looks like they were both correct
To determine who is correct between Austin and Tamara, we need to compare their completions of the square with the original quadratic equation.
The standard form of a quadratic equation is:
ax^2 + bx + c = 0
In this case, the equation is 0 = 3x^2 - 15x + 6.
To complete the square, we follow these steps:
Step 1: Take half of the coefficient of the x-term and square it.
a. In this case, the coefficient of the x-term is -15.
b. Half of -15 is -15/2, and (-15/2)^2 = 225/4.
Step 2: Add the result from Step 1 to both sides of the equation.
a. Adding 225/4 to both sides of the equation gives us:
0 + 225/4 = 3x^2 - 15x + 6 + 225/4
225/4 = 3x^2 - 15x + 225/4 + 24/4
225/4 = 3x^2 - 15x + 249/4
Step 3: Factor the quadratic to create a perfect square binomial.
a. Factoring the quadratic gives us:
0 = 3(x^2 - 5x + 3)
0 = 3(x^2 - 5x + 25/4 - 25/4 + 3)
0 = 3[(x - 5/2)^2 + 2/4]
0 = 3[(x - 5/2)^2 + 1/2]
Comparing this with Austin's completion of the square:
Austin: 0 = 3(x - 5/2)^2 - 51/4
We can see that Austin's completion matches up with the steps we followed. Therefore, Austin is correct.
On the other hand, Tamara's completion of the square doesn't match up.
Tamara: 0 = (x - 5/2)^2 - 17/4
Thus, Austin is the one who correctly completed the square.
To determine who is correct, we need to complete the square of the quadratic equation 0=3x^2 - 15x + 6 by following the proper steps.
1. First, factor out the leading coefficient 3 from the equation:
0 = 3(x^2 - 5x + 2)
2. To complete the square, take half of the coefficient of the x-term (-5/2), square it (25/4), and add it to both sides of the equation:
0 + 25/4 = 3(x^2 - 5x + 2 + 25/4)
3. Simplify the expression inside the parentheses:
25/4 = 3(x^2 - 5x + 33/4)
4. To write the equation in completed square form, factor the trinomial inside the parentheses as a perfect square:
25/4 = 3(x^2 - 5x + 25/4)
5. Rewrite the equation with the perfect square:
25/4 = 3(x - 5/2)^2
Comparing this result with Austin and Tamara's answers:
Austin: 0 = 3(x - 5/2)^2 - 51/4
Tamara: 0 = (x - 5/2)^2 - 17/4
We can see that Tamara's answer matches the result we obtained, while Austin's answer does not.
Therefore, Tamara is correct in completing the square of the quadratic equation 0 = 3x^2 - 15x + 6.