find the constants a, b and c such that or all values of x,
3x^2 - 5x + 1 = a(x+b)^2 +c
also whats the coordinate of the minimum point??
3x^2-5*x+1=a*(x^2+2*b*x+b^2)+c
Becouse):
(x+b)^2=x^2+2*b*x+b^2
3x^2-5*x+1=a*x^2+2*a*b*x+a*b^2+c
3x^2-5*x+1=
=a*x^2+(2*a*b)*x+(a*b^2+c)
a*x^2=3*x^2 Divide with x^2
a=3
(2*a*b)*x=-5x Divide with x
2*a*b=-5
2*3*b=-5
6b=-5 Divide with 6
b=(-5/6)
a*b^2+c=1
3*(-5/6)^2+c=1
(3*25/36)+c=1
(25/12)+c=1
(25/12)+c=1
c=1-(25/12)=(12/12)-(25/12)=(-13/12)
c=(-13/12)
a=3 , b=(-5/6) , c=(-13/12)
Function have extreme where first derivate (dy/dx)=0
d(3x^2-5*x+1)/dx
=2*3*x-5
=6*x-5
(dy/dx)=6*x-5
6*x-5=0
6*x=5 Divide with 6
x=(5/6)
y=3*(5/6)^2-5*(5/6)+1
=3*(25/36)-(25/6)-1
=(25/12)-(25/6)-1
=(25/12)-(50/12)-(12/12)=(-37/12)
Coordinates of extreme point:
(5/6 , -37/12)
Extreme point is minimum if secod derivate >0
Secon derivate (d^2y/dx^2) is derivate of first derivate.
In this case:
(d^2y/dx^2)=6>0
That is minimum.
To find the constants a, b, and c such that the equation 3x^2 - 5x + 1 = a(x + b)^2 + c holds for all values of x, we can compare the coefficients of the corresponding terms.
First, let's expand the right side of the equation:
a(x + b)^2 + c = a(x^2 + 2bx + b^2) + c
= ax^2 + 2abx + ab^2 + c
Now, compare the coefficients of the terms on both sides of the equation.
For the coefficient of x^2:
On the left side, the coefficient is 3.
On the right side, the coefficient is a.
Therefore, we have 3 = a.
For the coefficient of x:
On the left side, the coefficient is -5.
On the right side, the coefficient is 2ab.
Therefore, we have -5 = 2ab.
For the constant term:
On the left side, the constant term is 1.
On the right side, the constant term is ab^2 + c.
Therefore, we have 1 = ab^2 + c.
From the equation 3 = a, we can substitute a = 3 into the equation -5 = 2ab:
-5 = 2(3)b
-5 = 6b
b = -5/6
Substituting a = 3 and b = -5/6 into the equation 1 = ab^2 + c:
1 = 3(-5/6)^2 + c
1 = 3(25/36) + c
1 = 25/12 + c
c = 1 - 25/12
c = 12/12 - 25/12
c = -13/12
Therefore, the constants are:
a = 3
b = -5/6
c = -13/12
To find the coordinates of the minimum point, we need to determine the x-coordinate and the corresponding y-coordinate.
For the equation 3x^2 - 5x + 1, the x-coordinate of the minimum point can be found using the formula:
x = -b / (2a)
In this case, a = 3 and b = -5, so:
x = -(-5) / (2 * 3)
x = 5/6
To find the y-coordinate, substitute the value of x back into the original equation:
y = 3(5/6)^2 - 5(5/6) + 1
y = 3(25/36) - 25/6 + 1
y = 25/12 - 50/12 + 12/12
y = -13/12
Therefore, the coordinate of the minimum point is (5/6, -13/12).