1. Completing the Square Completing the square is a method to solve quadratic equations or rewrite them in vertex

1. Completing the Square

Completing the square is a method to solve quadratic equations or rewrite them in vertex form: y=a(x−h)2+ky = a(x-h)^2 + ky=a(x−h)2+k.

Steps:

  1. Start with a quadratic equation in standard form:
    ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 (if a≠1a \neq 1a=1, divide through by aaa first).

  2. Move the constant term (ccc) to the other side:
    ax2+bx=−cax^2 + bx = -cax2+bx=−c.

  3. Make the coefficient of x2x^2x2 equal to 1 by dividing through by aaa (if needed).

  4. Find the value to complete the square:
    Take half the coefficient of xxx (i.e., b2\frac{b}{2}2b​), square it, and add it to both sides.

  5. Rewrite the left-hand side as a perfect square trinomial and simplify the right-hand side.

  6. Solve for xxx by isolating it (take the square root and solve for xxx).

Example:

Solve x2+6x−7=0x^2 + 6x - 7 = 0x2+6x−7=0 by completing the square:

  1. Move −7-7−7:
    x2+6x=7x^2 + 6x = 7x2+6x=7.

  2. Complete the square:
    (62)2=9\left(\frac{6}{2}\right)^2 = 9(26​)2=9. Add 999 to both sides:
    x2+6x+9=16x^2 + 6x + 9 = 16x2+6x+9=16.

  3. Rewrite as a square:
    (x+3)2=16(x + 3)^2 = 16(x+3)2=16.

  4. Solve:
    x+3=±4x + 3 = \pm4x+3=±4.
    x=−3+4=1x = -3 + 4 = 1x=−3+4=1 or x=−3−4=−7x = -3 - 4 = -7x=−3−4=−7.

2. The Discriminant

The discriminant is a part of the quadratic formula and determines the nature of the roots of a quadratic equation.

Formula:

For ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0:
Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac

Interpretation of Δ\DeltaΔ:

  • Δ>0\Delta > 0Δ>0: Two distinct real roots.

  • Δ=0\Delta = 0Δ=0: One repeated real root (the roots are equal).

  • Δ<0\Delta < 0Δ<0: Two complex roots (no real solutions).

Example:

For 2x2−4x+2=02x^2 - 4x + 2 = 02x2−4x+2=0:

  • a=2,b=−4,c=2a = 2, b = -4, c = 2a=2,b=−4,c=2.

  • Δ=(−4)2−4(2)(2)=16−16=0\Delta = (-4)^2 - 4(2)(2) = 16 - 16 = 0Δ=(−4)2−4(2)(2)=16−16=0.
    The equation has one repeated root.

3. The Quadratic Formula

The quadratic formula provides the exact solutions to any quadratic equation.

Formula:

For ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0:
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​

Steps:

  1. Identify aaa, bbb, and ccc from the quadratic equation.

  2. Substitute into the formula.

  3. Simplify to find the solutions.

Example:

Solve x2+3x−4=0x^2 + 3x - 4 = 0x2+3x−4=0 using the quadratic formula:

  1. a=1,b=3,c=−4a = 1, b = 3, c = -4a=1,b=3,c=−4.

  2. Substitute:
    x=−3±32−4(1)(−4)2(1)x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)}x=2(1)−3±32−4(1)(−4)​​
    x=−3±9+162x = \frac{-3 \pm \sqrt{9 + 16}}{2}x=2−3±9+16​​
    x=−3±252x = \frac{-3 \pm \sqrt{25}}{2}x=2−3±25​​
    x=−3±52x = \frac{-3 \pm 5}{2}x=2−3±5​.

  3. Solve:
    x=−3+52=1x = \frac{-3 + 5}{2} = 1x=2−3+5​=1 or x=−3−52=−4x = \frac{-3 - 5}{2} = -4x=2−3−5​=−4.

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VERTEX AND EVERYTHING

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Here’s a comprehensive guide to finding the vertex, understanding the axis of symmetry, and determining the key features of a quadratic equation.

Quadratic Equation: Key Features

The standard form of a quadratic equation is:
y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c

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

  • (h,k)(h, k)(h,k) is the vertex.

  • aaa determines the direction of the parabola (opens upward if a>0a > 0a>0, downward if a<0a < 0a<0).

Finding the Vertex

You can find the vertex in two ways:

1. By Completing the Square

Convert y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c into vertex form y=a(x−h)2+ky = a(x-h)^2 + ky=a(x−h)2+k.

Example:
Find the vertex of y=x2+4x+1y = x^2 + 4x + 1y=x2+4x+1.

  1. Group x2x^2x2 and xxx: y=(x2+4x)+1y = (x^2 + 4x) + 1y=(x2+4x)+1.

  2. Complete the square:

    • Take half the coefficient of xxx, square it: (42)2=4\left(\frac{4}{2}\right)^2 = 4(24​)2=4.

    • Add and subtract 444: y=(x2+4x+4−4)+1y = (x^2 + 4x + 4 - 4) + 1y=(x2+4x+4−4)+1.

  3. Rewrite:
    y=(x+2)2−3y = (x+2)^2 - 3y=(x+2)2−3.
    Vertex: (−2,−3)(-2, -3)(−2,−3).

2. Using the Vertex Formula

The x-coordinate of the vertex is:
x=−b2ax = \frac{-b}{2a}x=2a−b​

To find the y-coordinate, substitute x=−b2ax = \frac{-b}{2a}x=2a−b​ into the equation.

Example:
Find the vertex of y=2x2−8x+5y = 2x^2 - 8x + 5y=2x2−8x+5.

  1. x=−(−8)2(2)=84=2x = \frac{-(-8)}{2(2)} = \frac{8}{4} = 2x=2(2)−(−8)​=48​=2.

  2. Substitute x=2x = 2x=2 into the equation:
    y=2(2)2−8(2)+5=8−16+5=−3y = 2(2)^2 - 8(2) + 5 = 8 - 16 + 5 = -3y=2(2)2−8(2)+5=8−16+5=−3.
    Vertex: (2,−3)(2, -3)(2,−3).

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex:
x=hx = hx=h (the x-coordinate of the vertex).

Example:

For y=x2+4x+1y = x^2 + 4x + 1y=x2+4x+1, the vertex is (−2,−3)(-2, -3)(−2,−3), so the axis of symmetry is:
x=−2x = -2x=−2.

Y-Intercept

The y-intercept is the point where the parabola crosses the y-axis (x=0x = 0x=0):
y=cy = cy=c.

Example:

For y=x2+4x+1y = x^2 + 4x + 1y=x2+4x+1:

  • Substitute x=0x = 0x=0: y=1y = 1y=1.

  • Y-intercept: (0,1)(0, 1)(0,1).

X-Intercepts (Roots or Zeros)

The x-intercepts are the points where the parabola crosses the x-axis (y=0y = 0y=0).
Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 using:

  1. Factoring (if possible).

  2. Completing the Square.

  3. The Quadratic Formula:
    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​.

Example:

For y=x2+4x+1y = x^2 + 4x + 1y=x2+4x+1, solve x2+4x+1=0x^2 + 4x + 1 = 0x2+4x+1=0:

  • Discriminant: Δ=42−4(1)(1)=16−4=12\Delta = 4^2 - 4(1)(1) = 16 - 4 = 12Δ=42−4(1)(1)=16−4=12.

  • Roots:
    x=−4±122=−4±232=−2±3x = \frac{-4 \pm \sqrt{12}}{2} = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3}x=2−4±12​​=2−4±23​​=−2±3​.

  • X-intercepts: x=−2+3,x=−2−3x = -2 + \sqrt{3}, x = -2 - \sqrt{3}x=−2+3​,x=−2−3​.

Graphing Summary

  1. Vertex: Locate using −b2a\frac{-b}{2a}2a−b​ or completing the square.

  2. Axis of Symmetry: Vertical line x=hx = hx=h.

  3. Y-Intercept: Point (0,c)(0, c)(0,c).

  4. X-Intercepts: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

  5. Direction of Opening: Upward if a>0a > 0a>0, downward if a<0a < 0a<0.

  6. Additional Points: Choose values of xxx around the vertex for symmetry.