Parabolas and Completing the Square

Conic Sections: Parabolas

General Form

The general form of a parabola is given by: y=ax2+bx+cy = ax^2 + bx + c

Vertex Form

The vertex form of a parabola is:
y=a(x−h)2+ky = a(x - h)^2 + k

In this form, the vertex of the parabola is at the point (h,k)(h, k).

Completing the Square

Completing the square is a method to convert the general form of a quadratic equation into vertex form.

Simple Example

Given: x2+4x+5x^2 + 4x + 5

  1. Recognize that x2+4xx^2 + 4x is part of (x+2)2(x + 2)^2.

  2. Rewrite: (x+2)2+1(x + 2)^2 + 1 because (x+2)2=x2+4x+4(x+2)^2 = x^2 + 4x + 4, and we have x2+4x+5x^2 + 4x + 5.

  3. Here, h=−2h = -2 and k=1k = 1, so the vertex is (−2,1)(-2, 1).

Advanced Example

Given: 3x2−7x+43x^2 - 7x + 4

  1. Divide by the coefficient of x2x^2 to get: 3(x2−73x)+43(x^2 - \frac{7}{3}x) + 4

  2. Identify the form x2−2xy+y2=(x−y)2x^2 - 2xy + y^2 = (x - y)^2.

  3. Determine yy such that 2xy=73x2xy = \frac{7}{3}x. Thus, 2y=732y = \frac{7}{3}, so y=−76y = -\frac{7}{6}.

  4. Add and subtract (76)2=4936(\frac{7}{6})^2 = \frac{49}{36} inside the parenthesis to complete the square: 3(x2−73x+4936)+4−3(4936)3(x^2 - \frac{7}{3}x + \frac{49}{36}) + 4 - 3(\frac{49}{36})

  5. Rewrite as: 3(x−76)2+4−49123(x - \frac{7}{6})^2 + 4 - \frac{49}{12}

  6. Simplify to: 3(x−76)2−1123(x - \frac{7}{6})^2 - \frac{1}{12}

  7. The vertex is therefore (76,−112)(\frac{7}{6}, -\frac{1}{12}).

Steps for Completing the Square

  1. Divide by the coefficient aa in front of x2x^2.

Proving the Quadratic Equation by Completing the Square

  1. Start with the general quadratic equation: ax2+bx+c=0ax^2 + bx + c = 0

  2. Divide by aa: x2+bax+ca=0x^2 + \frac{b}{a}x + \frac{c}{a} = 0

  3. Move the constant term to the right side: x2+bax=−cax^2 + \frac{b}{a}x = -\frac{c}{a}

  4. To complete the square, we need to add and subtract (b2a)2(\frac{b}{2a})^2:

    • Identify 2xy=bax2xy = \frac{b}{a}x, then y=b2ay = \frac{b}{2a}, and y2=b24a2y^2 = \frac{b^2}{4a^2}.

  5. Add (b2a)2(\frac{b}{2a})^2 to both sides: x2+bax+b24a2=−ca+b24a2x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}

  6. Rewrite the left side as a square: (x+b2a)2=b24a2−ca(x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{c}{a}

  7. Find a common denominator on the right side: (x+b2a)2=b2−4ac4a2(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}