Math1000 section1.5
- 1. Math 1000 Stuart Jones Section 1.5 Equations
- 2. Math 1000 Stuart Jones In this section, we will look at a wide variety of different types of equations. Each of these will wiggle their way into your mathematical future. The first we will look at is a linear equation. Linear equations can have one solution, no solution, or ALL REAL NUMBERS as solutions. Let’s look at a few.
- 3. Math 1000 Stuart Jones Solve 3x + 4 = 16
- 4. Math 1000 Stuart Jones Solve −2(3x + 4) = 2(x − 4) − 8x
- 5. Math 1000 Stuart Jones Solve −2x + 7 = 2(−x − 5)
- 6. Math 1000 Stuart Jones The second type of equations we will look at are sometimes called literal equations. In this case, we are asked to solve a given equation for a certain variable.
- 7. Math 1000 Stuart Jones Solve V = πr2h for r
- 8. Math 1000 Stuart Jones Solve W = 4vr2s 3p for p
- 9. Math 1000 Stuart Jones The nest group of equations, quadratic equations, is a rather large group. We will study 3 different methods of solving them. The first, and sometimes easiest, is by factoring. To use this, we also need an added tidbit: Zero Product Property ab = 0 ⇐⇒ a = 0 or b = 0
- 10. Math 1000 Stuart Jones Solve by factoring: x2 + 2x − 63 = 0
- 11. Math 1000 Stuart Jones Solve by factoring: x2 − 81 = 0
- 12. Math 1000 Stuart Jones Completing the Square Start with a quadratic equation ax2 + bx + c = 0. 1 If the leading coefficient is not 1, divide both sides by a, the leading coefficient, to make it 1. 2 Move the constant term to the right side. 3 Take the b term (whatever it is after dividing by a), and divide it by 2. Remember this number. 4 Take the number from the previous step, and square it. 5 Add the number from the previous step to both sides of the equation. 6 Rewrite the left side of the equation as (x + @)2, where @ is the number that you divided by 2 above. 7 Simplify the right side, then square root both sides and move the constant to get x by itself.
- 13. Math 1000 Stuart Jones Solve by Completing the Square: x2 + 6x − 55 = 0
- 14. Math 1000 Stuart Jones Solve by Completing the Square: x2 + 6x − 55 = 0 For time’s sake, this is the only example we’ll do in the notes, but please do some of the homework problems on this! See me for help on this.
- 15. Math 1000 Stuart Jones Finally, with quadratic equations, we have the Quadratic Formula: Quadratic Formula If we have ax2 + bx + c = 0, then: x = −b ± √ b2 − 4ac 2a
- 16. Math 1000 Stuart Jones Solve by the Quadratic Formula: x2 − 6x + 2 = 0
- 17. Math 1000 Stuart Jones The next type of equation we will look at is a rational equation. The general strategy here is going to be to multiply to get rid of our denominators first.
- 18. Math 1000 Stuart Jones Solve: x x − 1 + x x + 1 = 0
- 19. Math 1000 Stuart Jones The last type of equation we will look at this section is radical equations. Here, the goal is to get a radical by itself, then get rid of it, and if needed, repeat. MAKE SURE to check your answer here, because it’s possible you could get fake answers (called extraneous) doing this.
- 20. Math 1000 Stuart Jones Solve: 2x + √ x + 1 = 1
- 21. Math 1000 Stuart Jones The Bottom Line This is a big section to highlight two points you’ll see throughout this class - MEMORIZE the formulas/procedures (they will not be given to you on the test), and practice! Do the homework problems. If you do not, you will likely struggle with this.