2. In solving quadratic equations, there are three ways in which to solve for x: - by square roots (directly) - by factoring - by completing the square - by the quadratic formula **Note: in this presentation, we assume all quadratic equations are monic, meaning . If this is not the case, divide all terms by “a” to make it monic.
6. -by square roots -by factoring -by completing the square -by the quadratic formula Isolate the x term….
7. -by square roots -by factoring -by completing the square -by the quadratic formula Square root both sides….
8. -by square roots -by factoring -by completing the square -by the quadratic formula Remember the plus and minus….
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10. With the numbers from the first step, determine if any combination of the numbers when added together result in “b”.
11. These numbers from the second step (if it works) now need to be written as (x-(first number) )x(x-(second number))=0
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13. -by square roots -by factoring -by completing the square -by the quadratic formula Determine the numbers that multiply together to be equal to -6:
14. -by square roots -by factoring -by completing the square -by the quadratic formula Now look at the sum of each of these pairs of numbers. If one pair is equal to -1=b, those are the numbers used to factor the expression
15. -by square roots -by factoring -by completing the square -by the quadratic formula Having identified our factors, we can write the above expression as….
16. -by square roots -by factoring -by completing the square -by the quadratic formula Lastly, we set each part (factor) equal to zero and solve for x:
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18. Identify the “bx” term in your equation and add to both sides of the equation.
21. -by square roots -by factoring -by completing the square -by the quadratic formula First we move the x terms to one side of the equation and the other terms to the other side of the equation….
22. -by square roots -by factoring -by completing the square -by the quadratic formula Next, we determine our b value and compute And add it to each side of the equation. So,
23. -by square roots -by factoring -by completing the square -by the quadratic formula After this, the left side of the equation will factor (this is left up to you to verify) as
24. -by square roots -by factoring -by completing the square -by the quadratic formula Lastly, finish solving for x using square roots and other basic algebra skills:
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26. Substitute in the appropriate values into the following formula and simplify to obtain your solutions:(This is the quadratic formula)
27. -by square roots -by factoring -by completing the square -by the quadratic formula This problem can be solved by completing the square, but the quadratic formula is going to be used to determine the x values.
28. -by square roots -by factoring -by completing the square -by the quadratic formula First we determine our a, b, and c:
29. -by square roots -by factoring -by completing the square -by the quadratic formula Then we use the quadratic formula with these values: Thus
