f(x) = ax^2 + bx + c --> f(x) = a(x-b)^2 + c
In the completed-square form, (b, c) are the co-ordinates of the vertex (which is the maximum/minimum point). So for part a:
- Here is the original
f(x) = x^2 + 12x + 11
- Halve the x-coefficient (the number before x) and use it as -b in the vertex form described above. You must then calculate the square of this number and minus it at the end because when you multiply it out, this is the surplus you will make along with x^2 + 12x:
f(x) = (x + 6)^2 - 36 + 11
- Tidy this up by collecting the constant (just number) terms together:
f(x) = (x + 6)^2 - 25
- Using the form a(x - b)^2 + c, we can work out that b = -6 (because -b = 6) and c = -25. This gives us the vertex (-6, -25) which is a minimum point because the graph is a positive quadratic, giving us the characteristing 'U' shape which has a bottom. If it were a negative quadratic (denoted by a negative x^2 coefficient), the vertex will be a maximum point because it has an 'n' shape instead.
- To solve the equation from here, make the function equal to zero:
(x + 6)^2 - 25 = 0
- Then take 25 to the other side:
(x + 6)^2 = 25
- Next, square-root both sides:
x + 6 = ±5
- Rearrange to finish:
x = -6 ± 5 = -1 or -11
- Therefore, the roots (solutions) to the equation x^2 + 12x + 11 = 0 are x = -1 or -11
This method will work the same for the other equations up there too, so I will leave them for you to do.
I hope this helps
b) (-6, -25)
Perfect square trinomial! If you have 12x, divide by two and get 6, so we have (x+6)^2+q. Then solve for q. You get q as -25.
For b, we just take the vertex from vertex form. We get x+6=0 so x=-6, and q = -25, so the answer is (-6, -25).
Hope that helped,
A) 1. Factor x^2-2x-24
3. Set the factors equal to 0. x-6=0, and x+4=0.
4. Solve for x. x-6+6=0+6, x=6. x+4-4=0-4, x=-4.
5. Solutions are x= 6, -4
1. Factor x^2-11x+18
3. Set factors equal to 0. x-9=0, and x-2=0.
4. Solve for x. x-9+9=0+9, x=9. x-2+2=0+2, x=2
5. Solutions are x=9, 2.
On trying to factorize the given trinomial we could evaluate that there are no like terms. Therefore, the trinomial itself is the factor.
The above solution is same in this case. On trying to factorize the given trinomial we could evaluate that there are no like terms. Therefore, the trinomial itself is the factor.
Steps to solve a trinomial:
A trinomial is a polynomial consisting of three terms or monomials. The following are the steps to be followed on factorizing a trinomial,
Step 1: Identify a, b , and c in the trinomial
Step 2: Write down all factor pairs of c.
Step 3: Identify which factor pair from the previous step sum up to b.
Step 4: Substitute factor pairs into two binomials.
distributive property is a(b+c)=ab+ac
answer is B
x² + 2x
x(x + 2)
x² + 2x = 0
x(x + 2) = 0
x = 0 or x + 2 = 0
x = 0 or x = - 2
To get -2 from 1 you minus 3
So the answer in complete the square format would be (x+1)^2 -3
B) x1=2 x2=9
40% off the regular price is the correct answer.