Prerequisites

1
Prerequisites
2
Solving Equations Graphically, Numerically, and
Algebraically

• P.5

3
Solving Quadratic Equations

• Graphically
• Factoring
• Extracting Square Roots
• Completing the Square
• Quadratic Formula
• Intersections on a graph
• Tables

4
Graphically

• Rewrite the equation so that 0 is on one side and
  all other terms are on the other.
• Place the equation in y1.
• Graph
• Use the zeros function on the calculator to find
  the solutions to the equation or the x-intercepts.

5
Graphically
6
Graphically
7
Factoring
8
Factoring--Quadratic

• Rewrite the equation so that 0 is on one side and
  all other terms are on the other.
• Combine like terms.
• Factor.
• Set each factor to zero and solve.

9
Extracting Square roots

• Solve the equation for the ( )2
• Take the square root of both sides. Dont forget
  the plus and minus.
• Solve the result for x.

10
Extracting Square roots

• (2x 3)2 169
• The equation has been solved for the squared
  term.
• Take the square root of both sides and remember
  the .
• 2x 3 13
• Subtract three from both sides.
• 2x 10
• Divide both sides by 2.
• X 5

11
Completing the Square

• Rewrite and simplify the equation so that all the
  variables are on the left and the constant is on
  the right.
• If the coefficient of the x2 term is not one,
  then divide all terms by the coefficient.
• Take the coefficient of the x term and divide it
  by two and square the result. Add this number to
  both sides of the equation.
• Factor the left side of the equation.
• Solve by extracting the square root.

12
Completing the Square

• 3x2 - 6x 7 x2 3x x(x1) 3
• Distribute.
• 3x2 - 6x 7 x2 3x x2 - x 3
• Combine like terms on each side.
• 3x2 - 6x 7 2x 3
• Subtract 2x and add 7 from/to each side.
• 3x2 8x 10

13
Completing the Square

• 3x2 8x 10
• Divide both sides by 3.
• x2 (8/3)x 10/3
• Take the coefficient of the x and divide it by 2
  and then square it.
• (-8/3)/2 -8/6 -4/3
• (-4/3)(-4/3) 16/9
• Add this number to both sides of the equation.
• x2 (8/3)x 16/9 10/3 16/9 46/9
• Factor the left side.
• (x - 4/3)2 46/9

14
Completing the Square

• (x - 4/3)2 46/9
• Take the square root of both sides.
• X 4/3 sqrt(46)/3
• Add 4/3 to both sides.
• X sqrt(46)/3 4/3

15
Quadratic Formula

• Rewrite the equation so that zero is on one side
  and all other terms are on the other.
• Combine like terms.
• Determine a, b, and c.
• Use the quadratic formula to find the possible x
  values.

16
Use a Graph to Solve

• Estimate where the graph crosses the x-axis for x
  intercepts.
• Estimate where the graph crosses the y-axis for y
  intercepts.
• X-intercepts are solutions to a quadratic
  equation in x.

17
Use a Graph to Solve
Find the x-intercepts. In this case they are 1
and 3. Remember that x intercepts are the spots
where y 0 or solutions to quadratic
equations. X 1 or x 3
18
Solutions from a Table

• The zeros or solutions to a quadratic equation
  will lie between positive and negative values for
  y.
• As y travels from positive to negative, as long
  as the graph has no holes or jumps, the graph
  will have to pass through zero.