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Solving Quadratics: Methods & Examples by Professor Owl

Learn how to solve quadratic equations by graphing, factoring, completing the square, and using the quadratic formula. Understand key concepts such as discriminant and vertex. Practice with examples provided by Professor Owl.

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Solving Quadratics: Methods & Examples by Professor Owl

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  1. Solving Quadratics Equations With Professor Owl Created by Robbie Smith

  2. Solve by Graphing Quadratic Term: ax² Linear Term: bx Constant Term: c In order to have a solution, the line or parabola must touch the x-axis once or twice. If it doesn’t touch at all, there is no solution. You must also find the vertex and axis of symmetry

  3. Solve by Graphing • To find the points, you can make a table! Then graph the results. Ex. x²+6x+14=y Vertex: (-3,5) Axis or Sym: -3 Solution: None This graph shows that there is no solution.

  4. Solve by Factoring • Remember: Most equations can be factored, but not all equations can be factored. • Here are some examples of factoring. Unlike the last example which was already factored, you must factor this one. Ex. 16x² -9=0 (4x+3)(4x-3)=0 4x+3=0 4x-3=0 -3 -3 +3 +3 4x=-34x=3 4 4 4 X=-3 x=3 4 4 Ex. (x+5)(3x-2)=0 X+5=0 3x-2=0 -5 -5 +2 +2 X=-5 3x=2 3 3 x=2 3

  5. Solve by Factoring • More Examples Set the equation equal to zero Ex. x²+16=8x -8x -8x x²-8x+16=0 (x-4)(x-4)=0 x-4=0 +4 +4 x=4 Write in Proper Form Ex. 8x² -2=-2x +2x +2x 8x² -2=0 x²-16=0 (8x+4)(8x-4) 8 (x+4)(8x-4) x=-4 x=1 2 Ex. 3x²+6x=0 3x(x+2)=0 x= 0 x= -2

  6. Completing the Square • Examples of Perfect Squares x²+4x+4 (x+2)(x+2)=(x+2)² x²-8+16 (x-4)(x-4)=(x-4)² x²+14x+49 (x+7)(x+7)=(x+7)² To find the constant, do the following: Divide Linear by 2 Then square it. x²+4x+? 4/2=2 2²=4 x²+4x+4

  7. Completing the Square • Examples x²+4-10=0 +10 +10 x²+4x=10 4/2=2 2²=4 x²+4x+4=10+4 (x+2)²=14 √(x+2)²=√14 x+2=√14 x=-2+√14 x=-2-√14 3x²+6x-9=0 3 3 3 x²+2x-3=0 x²+2x+1=3+1 √(x+1)=√4 x+1=2 x+1=-2 x=1 x=-3

  8. Solve by the Quadratic Formula • When you get an equation, it looks like this: ax²+bx+c When using the quadratic formula, use this formula: x=-b±√b²-4ac 2a The Discriminant: b²-4ac (Very Important) It tells you the numbers, root, and solutions. Sweet! Let’s see an example!

  9. Solve by the Quadratic Formula x=-6±12 2(3) x=-6+12 6 x=-6-12 6 3x²+6x-9=0 6²-4(3)(-9) 36+108=144 144: Two Reals Rational x=1 x=-3 You must find the discriminant! Discriminant: Negative-2 Imaginary Solutions Zero- 1 Real Solution Positive-perfect Square- 2 Reals Rational Positive-Non-perfect square- 2 Reals Irrational

  10. And That’s It!

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