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Hawkes Learning Systems: Intermediate Algebra

Hawkes Learning Systems: Intermediate Algebra. Section 7.1a: Quadratic Equations: The Square Root Method. Objectives. Solve quadratic equations by factoring Solve quadratic equations by using the definition of a square root. . Solving Quadratic Equations by Factoring.

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Hawkes Learning Systems: Intermediate Algebra

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  1. Hawkes Learning Systems:Intermediate Algebra Section 7.1a: Quadratic Equations: The Square Root Method

  2. Objectives • Solve quadratic equations by factoring • Solve quadratic equations by using the definition of a square root.

  3. Solving Quadratic Equations by Factoring When the solutions of a polynomial equation can be found by factoring, the method depends on the zero-factor property. Zero-Factor Property If the product of two factors is 0, then one or both of the factors must be 0. For factors a and b, For example,

  4. Solving Quadratic Equations by Factoring An equation that can be written in the form Where a, b, and c are real numbers and is called a quadratic equation.

  5. To Solve an Equation by Factoring • Add or subtract terms so that one side of the equation is 0. • Factor the polynomial expression. • Set each factor equal to 0 and solve each of the resulting equations. Note: If two of the factors are the same, then the d solution is said to be a double root or a root d of multiplicity two.

  6. Example 1: Solving Quadratic Equations by Factoring Solve the following quadratic equation by factoring. Subtract 18 from both sides. One side must be zero. Factor the left-hand side. or Set each factor equal to 0. Solve each linear equation.

  7. Example 2: Solving Quadratic Equations by Factoring Solve the following quadratic equation by factoring. The solution is a double root.

  8. Solving Quadratic Equations by Factoring • Quadratic equations may have nonreal complex solutions. • In particular, the sum of two squares can be factored as the product of complex conjugates. • For example,

  9. Example 3: Quadratic Equations Involving the Sum of Two Squares Solve the following equation by factoring. Check:

  10. Square Root Property v If , then If, then or , or . Note: If c is negative, then the solutions will be nonreal.

  11. Using the Definition of Square Root and the Square Root Property Consider the equation Allowing that the variable, x, might be positive or negative, we use the definition of square root, Taking the square root of both sides of the equation gives: So we have two solutions,

  12. Using the Definition of Square Root and the Square Root Property Similarly, for the equation the definition of square root gives This leads to the two equations and the two solutions, as follows:

  13. Example 4: Using the Square Root Property Solve the following quadratic equations by using the Square Root Property. a. b.

  14. Hawkes Learning Systems:Intermediate Algebra Section 7.1b: Quadratic Equations: Completing the Square

  15. Objectives • Solve quadratic equations by completing the square • Find polynomials with given roots.

  16. Completing the Square • Recall that a perfect square trinomial is the result of squaring a binomial. • Our objective here is to find the third term of a perfect square trinomial when the first two terms are given. This is called completing the square.

  17. Completing the Square

  18. Completing the Square Step 1: Write the equation in the form . Step 2: Divide byif, so that the coefficient of is : . Step 3: Divide the coefficient of by , square the result, and add this to both sides. Step 4: The trinomial on the left side will now be a perfect square. That is, it can be written as the square of an algebraic expression.

  19. Example 1: Completing the Square Add the constant that will complete the square for the expression. Then write the new expression as the square of a binomial. Solution: Since the leading coefficient is 1, we can begin to complete the square. We take half of the coefficient of the x term and square the result.

  20. Example 1: Completing the Square Adding 81 to the expression yields the perfect square trinomial This can be factored as Thus completing the square gives us:

  21. Example 2: Completing the Square Solve the following quadratic equation by completing the square. Take half of the coefficient of the x term. Square the result.

  22. Example 2: Completing the Square (Cont.) Adding 9 to both sides of the original equation will result in a perfect square trinomial. Add 9 to both sides (completing the square). Factor the left-hand side. Use the Square Root Property to solve. or or We write these two solutions as:

  23. Example 3: Completing the Square Solve the following quadratic equation by completing the square. Divide each term by 2 so that the leading coefficient will be 1. Isolate the constant term. Take half of the coefficient of the x term, and square it.

  24. Example 3: Completing the Square (Cont.) Add 4 to both sides of Factor the left-hand side of the equation. Use the Square Root Property. or or

  25. Writing Equations with Known Roots To find the quadratic equation that has the given roots and , first get 0 on one side of each equation. Set the product of the two factors equal to 0 and simplify. Continued on the next slide…

  26. Writing Equations with Known Roots Regroup the terms to present the product of complex conjugates. This makes the multiplication easier.

  27. Example 4: Equations with Known Roots Find the quadratic equation that has the given roots. and Get 0 on one side of each equation. Set the product of the two factors equal to 0.

  28. Example 4: Equations with Known Roots (Cont.) Simplify and solve.

  29. Hawkes Learning SystemsIntermediate Algebra Section 7.2: Quadratic Equations: The Quadratic Formula

  30. Objectives • Write quadratic equations in standard form. • Identify the coefficients of quadratic equations in standard form. • Solve quadratic equations using the quadratic formula. • Determine the nature of the solutions (one real, two real, or two non-real) for quadratic equations using the discriminant.

  31. Standard Form of the Quadratic Equation We are interested in developing a formula that can solve quadratic equations of any form. This formula will always work, though other techniques may be more convenient to use. We want to solve the standard form equation of the quadratic formula for xin terms of a, b and c. The standard formof the quadratic equation is where , , and are real numbers and .

  32. Development of the Quadratic Formula Standard form. Add –c to both sides. Divide both sides by a so that the leading coefficient is 1. Complete the square. Simplify. Continued on the next page…

  33. Development of the Quadratic Formula Find LCM of the denominators on the right side. Combine the fractions on the right side of the equation. Square Root method. The quadratic formula.

  34. Development of the Quadratic Formula The quadratic formula, , can ALWAYS solve quadratic equations of any form. Because it is so useful, you should memorize the quadratic formula.

  35. Discriminant The expression is called the discriminant. The discriminant determines the number of solutions to the given quadratic equation. If the discriminant is:

  36. Solve Quadratic Equations Using the Quadratic Formula Ex: Solve the quadratic equation using the quadratic formula. Compare it to the standard quadratic equation to find a, b and c. Solve the discriminant by plugging a, b and c into . Since the discriminant is 0, there is one real solution. Use the quadratic formula to find the solution, x. We know that the discriminant is 0, so we can just plug that in. Solution: ̶ 1

  37. Example 1: Solve Quadratic Equation Using the Quadratic Formula Solve the quadratic equation using the quadratic formula. Discriminant: Number of solutions: Real or non-real: Compare it to the standard quadratic equation to find a, b and c. Solve the discriminant by plugging a, b and c into . What does it mean for a discriminant to be negative? Continued on the next page…

  38. Example 1: Solve Quadratic Equation Using the Quadratic Formula (Cont.) Solutions: Use the quadratic formula to find the solution, x. We know that the discriminant is ̶ 3, so we can plug that in.

  39. Example 2: Solve Quadratic Equation Using the Quadratic Formula Solve the quadratic equation using the quadratic formula. Discriminant: Number of solutions: Real or non-real: Compare it to the standard quadratic equation to find a, b and c. Solve the discriminant by plugging a, b and c into . What does it mean for a discriminant to be positive? Continued on the next page…

  40. Example 2: Solve Quadratic Equation Using the Quadratic Formula (Cont.) Solutions: Use the quadratic formula to find the solution, x. We know that the discriminant is 1, so we can plug that in.

  41. Hawkes Learning Systems:Intermediate Algebra Section 7.3: Applications- Quadratic Equations

  42. Objective Solve applied problems by using quadratic equations.

  43. Strategy for Solving Word Problems • Read the problem carefully. • Decide what is asked for and assign a variable to the unknown quantity. • Draw a diagram or set up a chart whenever possible. • Form an equation (or inequality) that relates the information provided. • Solve the equation or inequality. • Check your solution with the wording of the problem to be sure that it makes sense.

  44. The Pythagorean Theorem • Problems involving right triangles often require the use of quadratic equations. • In a right triangle, one of the angles is a right angle (measures ), and the side opposite this (the longest side) is called the hypotenuse. • The other two sides are called legs. hypotenuse leg leg

  45. The Pythagorean Theorem In a right triangle, the square of the hypotenuse ( ) is equal to the sum of the squares of the legs ( and ).

  46. Example 1: The Pythagorean Theorem The width of a rectangle is 5 yards less than its length. If one diagonal measures 25 yards, what are the dimensions of the rectangle? Solution: Draw a diagram for problems involving geometric figures whenever possible. Let Then, by the Pythagorean Theorem,

  47. Example 1: The Pythagorean Theorem (Cont.) Now, solve the quadratic equation. A negative number does not fit the conditions of the problem. Length: Width:

  48. Projectiles The formula is used in physics and relates to the height of a projectile such as a thrown ball, a bullet or a rocket. height of object, in feet. time object is in the air, in seconds. beginning velocity, in feet per second. beginning height. if object is initially at ground level.

  49. Example 2: Projectiles A bullet is fired straight up from 6 feet above ground level with a muzzle velocity of 420 ft per sec. When will the bullet hit the ground? Solution: The bullet hits the ground when Divide both sides by 2.

  50. Example 2: Projectiles Use the quadratic formula to solve the equation. Therefore, the bullet hits the ground in 26.26 seconds.

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