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1. 1. Solving Quadratic Equations<br />( x + 4) ( x + 6 )<br />( x – 7) ( x – 3 )<br />( x – 2) ( x + 8 )<br />Algebra Unit – Part 1<br />( x + 1) ( x - 5 )<br />
2. 2. Why did you learn to factor a trinomial?<br />You learned to factor a trinomial into TWO binomials in order to use those answers.<br />In order to use those answers you have to set each binomial = to zero (0).<br />( X – 3 ) ( x + 5 ) = 0 <br />If you multiply two binomials and the value of one of them is zero, then the whole product is zero.<br />
3. 3. Setting the binomial = to Zero<br />The product ( x – 3 ) ( x + 5 ) = 0<br />If either binomial = zero then the whole things is zero.<br />What number would make the 1st binomial = 0?<br />3<br />What number would make the 2nd binomial = 0?<br />-5<br />That means if x = 3 or x = -5, the whole problem is 0. Therefore, our answer is x = 3 or x = -5.<br />
4. 4. Sample Problems <br />In the middle of your sheet of notes fill in the answers as we go, by setting each binomial = to zero.<br />( x – 2 ) ( x + 8) = 0<br />X = 2 or x = -8<br />( x – 7 ) ( x – 3 ) = 0<br />X = 7 or x = 3<br />( x + 4 ) ( x + 6 ) = 0<br />X = -4 or x = -6<br />( x + 1 ) ( x – 5 ) = 0<br />X = -1 or x = 5<br />
5. 5. Solving Basic Square Root Problems<br />The easiest type of quadratics to solve is basic square root problems.<br />They come in two forms: <br />x2= 64 and x2 – 36 = 0<br />To solve the first one, all you do is take the square root of the number. X = √64 = 8 and -8<br />To solve the second one, you have to add 36 to both sides and then take the square root.<br />x2 – 36 = 0<br /> +36 +36<br /> x2 = 36<br />X = √36 = 6 and-6<br />
6. 6. Sample Problems<br />At the end of your sheet of notes fill in the answers as we go, by setting each binomial = to zero.<br /><ul><li>x2= 81
7. 7. X = √81 = 9 and-9
8. 8. x2= 9
9. 9. X = √9= 3 and-3
10. 10. x2 = 16
11. 11. X = √16 = 4 and-4</li></ul>x2 – 25 = 0<br />+25 +25<br /> x2 = 25<br />X = √25 = 5 and-5<br />x2 – 49 = 0<br />+49 +49<br /> x2 = 49<br />X = √49 = 7 and-7<br />x2 – 4= 0<br />+4 +4<br /> x2 = 4<br />X = √4= 2 and-2<br />
12. 12. Steps to solving regular quadratics<br />Set the trinomial = to zero.<br />Factor the trinomial into the product of binomials<br />Set each binomial = to zero<br />Solve for x<br />Example<br />Solve: x2+ 5x -24 = 0<br />Only way to get a negative at the end is multiply 1 positive & 1 negative, looking at middle number the bigger number needs to be positive.<br />( x + 8 ) ( x - 3) = 0<br />( x + 8 ) = 0 or ( x – 3) = 0<br />X = -8 or x = 3<br />
13. 13. Sample regular quadratic problems<br />In the middle of your sheet of notes fill in the answers as we go, by creating binomials and then setting each binomial = to zero.<br />x2+ 12x + 32= 0<br />( x + 8 ) ( x + 4) = 0<br />( x + 8 ) = 0 or ( x + 4) = 0<br />X = -8 or x = -4<br />X2- 8x + 15= 0<br />( x - 5 ) ( x - 3) = 0<br />( x - 5 ) = 0 or ( x - 3) = 0<br />X = 5or x = 3<br /><ul><li>X2- 8x - 9 = 0
14. 14. ( x - 9 ) ( x + 1) = 0
15. 15. ( x - 9 ) = 0 or ( x + 1) = 0
16. 16. X = 9 or x = -1</li></li></ul><li>Adding one more step<br />Again the first thing you need to do is set the trinomial = to zero. Therefore you may need to add or subtract in order to do so.<br />Example<br />x2+ 13x = -30<br /> +30 + 30<br />x2+ 13x + 30= 0<br />( x + 10 ) ( x + 3) = 0<br />( x + 10 ) = 0 or ( x + 3) = 0<br />X = -10 or x = -3<br />
17. 17. Sample one more step problems<br />At the end of your sheet of notes fill in the answers as we go, by setting the trinomials equal to zero, creating binomials and then setting each binomial = to zero.<br />Example 2<br />x2 + 2x = 63<br />-63 -63<br />x2+ 2x - 63= 0<br />( x + 9) ( x - 7) = 0<br />( x + 9 ) = 0 or ( x - 7) = 0<br />X = -9 or x = 7<br />Example 1<br />x2 + 8x = -12<br />+12 + 12<br />x2+ 8x + 12= 0<br />( x + 6) ( x + 2) = 0<br />( x + 6) = 0 or ( x + 2) = 0<br />X = -6or x = -2<br /><ul><li>Example 3
18. 18. x2 - 14x + 60 = 12
19. 19. -12 -12
20. 20. X2 -14x +48 = 0
21. 21. ( x - 8) ( x - 6) = 0
22. 22. ( x - 8 ) = 0 or ( x - 6) = 0
23. 23. X = 8 or x = 6</li>