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Question Find the minimum point of the curve y=x2−6x+5y = x^2 - 6x + 5 by completing the square.

Studdy Solution

STEP 1

Assumptions
1. The given quadratic equation is y=x2−6x+5y = x^2 - 6x + 5.
2. We are asked to find the coordinates of the minimum point of the curve, which is the vertex of the parabola represented by the quadratic equation.
3. Completing the square is a method used to rewrite a quadratic equation in vertex form, which is y=a(x−h)2+ky = a(x-h)^2 + k, where (h,k)(h,k) is the vertex of the parabola.

STEP 2

To complete the square, we need to express the quadratic equation in the form y=a(x−h)2+ky = a(x-h)^2 + k. We start by identifying the coefficient of x2x^2, which is aa, and the coefficient of xx, which is bb.
a=1,b=−6a = 1, \quad b = -6

STEP 3

We rewrite the quadratic equation, leaving a space to add and subtract a certain number that will allow us to complete the square.
y=x2−6x+□−□+5y = x^2 - 6x + \square - \square + 5

STEP 4

The number that we need to add and subtract is (b2)2\left(\frac{b}{2}\right)^2. This is derived from the binomial expansion (x−b2)2=x2−bx+(b2)2(x - \frac{b}{2})^2 = x^2 - bx + \left(\frac{b}{2}\right)^2.
(b2)2=(−62)2=(−3)2=9\left(\frac{b}{2}\right)^2 = \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9

STEP 5

Add and subtract the number calculated in STEP_4 inside the equation.
y=x2−6x+9−9+5y = x^2 - 6x + 9 - 9 + 5

STEP 6

Group the perfect square trinomial and the constants.
y=(x2−6x+9)−9+5y = (x^2 - 6x + 9) - 9 + 5

STEP 7

Recognize the perfect square trinomial as the square of a binomial.
y=(x−3)2−9+5y = (x - 3)^2 - 9 + 5

STEP 8

Combine the constants to simplify the equation.
y=(x−3)2−4y = (x - 3)^2 - 4

STEP 9

Now that we have the quadratic equation in vertex form, we can identify the vertex of the parabola, which gives us the coordinates of the minimum point.
The vertex form is y=(x−h)2+ky = (x - h)^2 + k, so comparing it with our equation y=(x−3)2−4y = (x - 3)^2 - 4, we get:
h=3,k=−4h = 3, \quad k = -4

STEP 10

The coordinates of the minimum point (vertex) of the curve are (h,k)(h, k).
Minimum point=(3,−4)\text{Minimum point} = (3, -4)
The coordinates of the minimum point of the curve y=x2−6x+5y = x^2 - 6x + 5 are (3,−4)(3, -4).

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