The vertex form is 2(x-1)^2+3.
To express the quadratic equation 2x^2 −4x+5=0 in vertex form, we need to complete the square. The vertex form of a quadratic equation is given by a(x−h)^2 +k, where (h,k) is the vertex of the parabola.
Starting with the given equation, 2x^2 −4x+5=0, first, factor out the coefficient of x^2 from the x^2 and x terms, which is 2 in this case:
2(x^2 −2x)+5=0.
Now, complete the square inside the parentheses. Take half of the coefficient of x (which is −2/2=−1), square it ((−1)^2 =1), and add it inside the parentheses:
2(x^2 −2x+1)+5−2=0.
Simplify the expression:.
2(x−1)^2 +3=0.
Now, the quadratic equation is in vertex form a(x−h)^2 +k, where a=2, h=1, and k=3. Therefore, the vertex form of the given quadratic equation is 2(x−1)^2+3=0. This form allows us to easily identify the vertex of the parabola, which is (1,3).