y=x^2-2x+3?

Question

answer for math questions about parabolas

Answer 1

x² - 2x + 3 = (x - 1)² + 2

                = (x - 1)² - (-2)

                = (x - 1 + sqrt(-2)) (x - 1 - sqrt(-2))

                = (x - 1 + sqrt(2) * i) (x - 1 - sqrt(2) * i)

Answer 2

Parabolas are of the form ax²+bx+c, where a,v and c are the coefficients (numbers) in front of the variable. Notice that all parabolas are of degree 2 (that means the highest power of the variable is 2).

There are three major methods of solving for the roots of a parabola. (roots being the places where the y value will be zero).

1. factor the tri-nomial (three terms) into two binomials (two terms).

2. complete the square (see below)

3. or use the quadratic formula -b + sqrt(b² - 4ac) then divide the whole thing by 2a

for the parabola stated above: y =

 x² - 2x + 3 factoing will not be the best way.

 

completing the square is done as follows:

set the equation = 0.  0 = x² -2x + 3

take the last term (3) and subtract it from both sides which gives you x² - 2x ⁼ -3

take the coefficient of the 'x' term which is -2, take half of it (-1) square it (1)

add this term (1) to both sides which gives you x² - 2x +1 = -2

NOTICE: you have made a perfect square of the left side which is (x-1)² so....

(x-1)² =-2.  Now take the sqrt of both sides which gives you x-1 =+sqrt(-2)

so..... the roots are complex! not real. they are x = 1-i*sqrt(2) and 1+i*sqrt(2)

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NOTE: The vertex or (maximum/minimum point) of the parabola ca be gotten by using the co-ordinate pair (-b/2a,f(-b/2a))

((-2/2),6) or ((-1,6)

 

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Using the quadratic formula we get -(-2) + sqrt(-2²-4(a)(3) all divided by (2*1)

which gives you 2 +  sqrt(4 - 12) all divided by 2

which gives you 2 + sqrt(-8) all divided by 2, which yields 2 +  2i* sqrt(2) alld divided by 2

whihc is 1 + i*sqrt(2) and 1 - i*sqrt(2)

same answer as completing the square!

Answer 3

Where are the qustions? Your parabola is opened up because the first coefficient is positive, The parabola has two roots defined by the equation

                                         x² -2x + 3 = 0

Factor it:

                                     x² -2x +3 = (x-3)(x+1) = 0

Thus the roots are   x₁ = 3   and x₂ = -1. The  axis of symmetry for the parabola is the midpoint of the segment [-1,3], which means x - coordinate of the vertex is  -1 + [3-(-1)]/2 =  1.  Because the parabola is opened up it has only minimum which is equal to y(1) = 2.