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Math Project 2ppt

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  • 1. Solving Equations and Inequalities Lindsay Lehman And Dana Bailey
  • 2. Linear Equations
    • 5x + 3 = 28
    • -3 -3
    • 5x = 25
    • ÷ 5 ÷5
    • x = 5
  • 3. Quadratic Equations
    • There are multiple ways to solve quadratic equations. These include:
      • Factoring
      • The quadratic formula
      • Completing the square
  • 4. Factoring
    • x^2 + 8x = -15
    • +15 +15
    • x^2 + 8x + 15 = 0
    • (x + 3)(x + 5) = 0
    • x + 3 = 0 x + 5 = 0
    • x = -3 and x = -5
    • Always plug your solutions back into the original equation to check for extraneous solutions.
  • 5. Using the Quadratic Formula
    • The quadratic formula is:
        • (-b) ± √(b^2)-4(a)(c)
        • 2(a)
    • When trying to solve quadratic equations and factoring doesn’t work, the alternative is to use the quadratic formula. It always works.
    • Simply take the parts of the equation and plug them into the formula. Then solve.
  • 6. Quadratic Formula Example
    • x^2 + 4x – 7 = 0
    • This equation does not factor, or at least not easily. Simply identify a, b, and c and plug them in to the quadratic formula.
    • For this equation:
      • A = 1
      • B = 4
      • C = -7
  • 7. Quadratic Example Continued
    • X = (-4)± √4^2 – 4(1)(-7)
    • 2(1)
    • Now just solve
    • (-4) + 2√11 = X (-4) – 2√11 = X
    • 2 2
  • 8. Completing the Square
    • Another alternative to use when a quadratic equation may not be factored easily is completing the square.
    • When you have an equation:
      • Ax^2 + Bx + C = 0
    • Just pull C to the opposite side (by subtracting or adding it).
    • Then, to make Ax^2 + Bx a perfect square, just add (b/2)^2 to both sides of the equation.
    • Once this is completed, write the equation in factored form, (x + b/2)^2 = c + (b/2)^2, then solve.
  • 9. Completing the Square Example
    • x^2 + 2x - 4 = 0
    • x^2 + 2x + __ = 4
    • x^2 + 2x + 1 = 4 + 1
    • (x + 1)^2 = 5
    • x + 1 = ± √5
    • x = -1 ± √5
  • 10. Rational Equations
    • Rational equations are basically fractions set equal to each other with variables in them.
    • All you have to do is get a common denominator (so that you can cancel them out), then solve the two expressions set equal to each other.
    • Or, if there is only a denominator on one side, you can multiply both sides by the common denominator to get rid of it, then solve.
  • 11. Rational Equation Example
    • 2x + 5 = 15
    • 4
    • 2x + 5 = (15x)4
    • 2x + 5 = 60
    • 2x = 55
    • x = 27.5
  • 12. Radical Equations
    • Radical Equations are equations that have radicals in them. They look tricky to deal with, but when you know how to handle them it’s not so bad.
  • 13. How to Solve Radical Equations
    • Solving radical is actually very simple.
    • Just follow these steps:
    • First, one must get the radical by itself on one side of the equation.
    • Then, one must square both sides (in order to eliminate the square root).
    • Once the square root is gone, simply solve the equation.
    • Finally, NEVER forget to check all solutions to weed out any extraneous ones.
  • 14. Radical Equation Example
    • x + 8 = √(5 + 2x) + 12
    • x + 8 = √(5 + 2x) + 12
    • -12 -12
    • (x – 4)^2 = (√(5 + 2x))^2
    • x^2 – 8x + 16 = 5 +2x
    • -(5 + 2x) -(5 + 2x)
    • x^2 – 10x + 11 = 0
  • 15. Radical Equation Ex. Continued
    • x = -(-10) ± √( 100 – 4(1)(11))
    • 2(1)
    • x = 10 ± √(56)
    • 2 2
    • = 5 ± √(14)
    • Solutions:
    • x ≈ 8.74, x ≈ 1.26
  • 16. Inequalities
    • Inequalities are equations with symbols in place of where the equals sign would be in a regular equation. There are four different signs, they are:
        • > - Greater than
        • < - Less than
        • ≤ - Less than or equal to
        • ≥ - Greater than or equal to
  • 17. Linear Inequalities
    • Solving a linear inequality is basically like solving a linear equation. The only difference is that when you divide or multiply by a negative, you have to flip the symbol around.
    • For example:
    • -2x + 5 > 15
    • -2x > 10
    • x < 5
  • 18. Quadratic Inequalities
    • Quadratic inequalities are basically solved the same way quadratic equations are solved.
    • For example:
    • x^2 + 5x + 6 > 0
    • (x + 2)(x + 3) > 0
    • x + 2 > 0 x + 3 > 0
    • Solutions:
    • x > -2 x > -3
  • 19. Solving by Graphing
    • Another way to solve any type of equation is to graph it. Generally, the easiest way to graph an equation/function is to first make a table. Then, just plot the points.
  • 20. Solving by Graphing Continued
    • Linear equations:
        • Just make a table to gather a few points, then graph them. It should look like a straight line. From the graph you can find both the x and y intercepts.
    • Quadratic equations:
        • Just like for linear equations, make a table and plot a few points. It would also be beneficial to find the vertex. By looking at the graph you can find the x and y intercepts and if it is a maximum or a minimum. Solving a quadratic equation by graphing means looking for the zeros.
  • 21. Solving by Graphing Continued
    • Radical Equations:
        • You can either make a table to gather several values or you can go from what you know about radical functions and just graph from there. (ie- √(x+5) just move to the left 5, and start curve on x-axis).
    • Rational Equations:
        • Again, you can either make a table or go off of what you already know about rational equations.
  • 22. Solving by Graphing Continued
    • Linear and Quadratic Inequalities:
        • First make a table and graph the line/parabola as you would for a linear/quadratic equation. To decide where to shade plug in the point (0, 0). If the point works in the equation then shade on the side of the line/parabola that the point is on. If it does not work when you plug it in, shade on the opposite side where that point is not.