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Chapter 3 Quadratic Functions and Equations

Chapter 3 Quadratic Functions and Equations. Quadratic Equations and Problem Solving. 3.2. Understand basic concepts about quadratic equations Use factoring, the square root property, completing the square, and the quadratic formula to solve quadratic equations Understand the discriminant

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Chapter 3 Quadratic Functions and Equations

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  1. Chapter 3 Quadratic Functions and Equations

  2. Quadratic Equations and Problem Solving 3.2 Understand basic concepts about quadratic equations Use factoring, the square root property, completing the square, and the quadratic formula to solve quadratic equations Understand the discriminant Solve problems involving quadratic equations

  3. Quadratic Equation A quadratic equation in one variable is an equation that can be written in the form ax2 + bx + c where a, b, and c are constants with a ≠ 0.

  4. Solving Quadratic Equations Quadratic equations can have no real solutions, one real solution, or two real solutions. The are four basic symbolic strategies in which quadratic equations can be solved. • Factoring • Square root property • Completing the square • Quadratic formula

  5. Factoring Factoring is a common technique used to solve equations. It is based on the zero-productproperty, which states that if ab = 0,then a = 0 or b = 0 or both. It is important toremember that this property works only for 0. For example, if ab = 1,then this equationdoes notimply that either a = 1 or b = 1.For example, a = 1/2 and b = 2 also satisfiesab = 1and neither a nor b is 1.

  6. Example: Solving quadratic equations with factoring Solve the quadratic equation 12t2 = t + 1. Check your results. Solution

  7. Example: Solving quadratic equations with factoring Solution continued Check:

  8. The Square Root Property Let k be a nonnegative number. Then the solutions to the equation x2 = k are given by

  9. Example: Using the square root property If a metal ball is dropped 100 feet from a water tower, its height h in feet above the groundafter tseconds is given by h(t)= 100 – 16t2.Determine how long it takes the ball to hitthe ground. Solution The ball strikes the ground when the equation100 – 16t2 = 0 is satisfied.

  10. Example: Using the square root property Solution continued The ball strikes the ground after 10/4, or 2.5, seconds.

  11. Completing the Square Another technique that can be used to solve a quadratic equation is completing the square.If a quadratic equation is written in the form x2+ kx=d, where k and d are constants,then the equation can be solved using

  12. Example: Completing the Square Solve 2x2 – 8x = 7. Solution Divide each side by 2:

  13. Symbolic, Numerical and Graphical Solutions Quadratic equations can besolved symbolically, numerically, and graphically. The following example illustrates eachtechnique for the equation x(x– 2)= 3.

  14. Symbolic Solution

  15. Numerical Solution

  16. Graphical Solution

  17. Quadratic Formula The solutions to the quadratic equation ax2 + bx + c = 0, where a ≠ 0,are given by

  18. Example: Using the quadratic formula Solve the equation 3x2 – 6x + 2 = 0. Solution Let a = 3, b = 6, and c = 2.

  19. The Discriminant If the quadratic equationax2+ bx+ c= 0 is solved graphically, the parabolay = ax2+ bx+ c can intersect thex-axis zero, one, or two times. Eachx-intercept is a real solution to the quadratic equation.

  20. Quadratic Equations and Discriminant To determine the number of real solutions to ax2 + bx + c = 0 with a ≠ 0, evaluatethe discriminant b2 – 4ac. 1.If b2 – 4ac > 0, there are two real solutions. 2.If b2 – 4ac = 0, there is one real solution. 3.If b2 – 4ac < 0, there are no real solutions.

  21. Example: Using the discriminant Use the discriminant to find the number of solutions to 4x2 – 12x + 9 = 0. Then solve the equation by using the quadratic formula. Support your answer graphically. Solution Let a = 4, b = –12, and c = 9 b2 – 4ac = (–12)2 – 4(4)(9) = 0 Discriminant is 0, there is one solution.

  22. Example: Using the discriminant Solution continued The only solution is 3/2.

  23. Example: Using the discriminant Solution continued The graph suggests there is only one intercept 3/2.

  24. Problem Solving Many types of applications involve quadratic equations. To solve these problems, we usethe steps for “Solving Application Problems” from Section 2.2.

  25. Example: Solving a construction problem A box is being constructed by cutting2-inch squares from the corners of a rectangular pieceof cardboard that is 6 inches longer than it is wide. If the boxis to have a volume of 224 cubic inches, find the dimensions of the piece of cardboard.

  26. Example: Solving a construction problem Solution Step 1: Let x be the width and x + 6 be the length. Step 2: Draw a picture.

  27. Example: Solving a construction problem Since the height times the width times the length must equal the volume, or 224 cubic inches, the following can be written 2(x – 4)(x + 2) = 224 Step 3: Write the quadratic equation in the form ax2 + bx + c = 0 and factor.

  28. Example: Solving a construction problem The dimensions can not be negative, so the width is 12 inches and the length is 6 inches more, or 18 inches. Step 4: After the 2-inch-square corners are cut out, the dimensions of the bottom of the box are 12 – 4 = 8 inches by 18 – 4 = 14 inches. The volume of the box is then 2•8•14 = 224 cubic inches, which checks.

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