1 / 15

Quadratic Equations P.7

Quadratic Equations P.7. A quadratic equation in x is an equation that can be written in the standard form ax 2 + bx + c = 0 where a , b , and c are real numbers with a not equal to 0. A quadratic equation in x is also called a second-degree polynomial equation in x.

pshumway
Download Presentation

Quadratic Equations P.7

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Quadratic EquationsP.7

  2. A quadratic equation in x is an equation that can be written in the standard form ax2+bx+c= 0 where a, b, and c are real numbers with a not equal to 0. A quadratic equation in x is also called a second-degree polynomial equation in x. Definition of a Quadratic Equation

  3. If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero. If AB= 0, then A= 0 or B= 0. The Zero-Product Principle

  4. Solving a Quadratic Equation by Factoring • If necessary, rewrite the equation in the form ax2+bx+c= 0, moving all terms to one side, thereby obtaining zero on the other side. • Factor. • Apply the zero-product principle, setting each factor equal to zero. • Solve the equations in step 3. • Check the solutions in the original equation.

  5. Text Example • Solve 2x2+ 7x= 4 by factoring and then using the zero-product principle. Step 1Move all terms to one side and obtain zero on the other side. Subtract 4 from both sides and write the equation in standard form. 2x2+ 7x- 4= 4 - 4 2x2+ 7x- 4 = 0 Step 2Factor. 2x2+ 7x- 4 = 0 (2x- 1)(x+ 4) = 0

  6. Solution cont. • Solve 2x2+ 7x= 4 by factoring and then using the zero-product principle. Steps 3 and 4Set each factor equal to zero and solve each resulting equation. 2 x- 1 = 0 or x+ 4 = 0 2 x= 1 x= -4 x = 1/2 Steps 5 check your solution

  7. (2x + -3)(2x + 1) = 5 4x2 - 4x - 3 = 5 4x2 - 4x - 8 = 0 4(x2-x-2)=0 4(x - 2)*(x + 1) = 0 x - 2 = 0, and x + 1 = 0 So x = 2, or -1 Example

  8. If u is an algebraic expression and d is a positive real number, then u2 = d has exactly two solutions. If u2 = d, then u = d or u = -d Equivalently, If u2 = d then u = d The Square Root Method

  9. If x2 + bx is a binomial then by adding (b/2) 2, which is the square of half the coefficient of x, a perfect square trinomial will result. That is, x2 + bx + (b/2)2 = (x + b/2)2 Completing the Square

  10. What term should be added to the binomial x2 + 8x so that it becomes a perfect square trinomial? Then write and factor the trinomial. The term that should be added is the square of half the coefficient of x. The coefficient of x is 8. Thus, (8/2)2 = 42. A perfect square trinomial is the result. x2 + 8x + 42 = x2 + 8x + 16 = (x + 4)2 Text Example

  11. B divided by 2, squared Don’t forget to add to both sides

  12. Quadratic Formula

  13. Discriminant b2 – 4ac Kinds of solutions to ax2+ bx+ c = 0 Graph of y= ax2+ bx+ c b2 – 4ac > 0 Two unequal real solutions Two x-intercepts b2 – 4ac = 0 One real solution (a repeated solution) One x-intercept b2 – 4ac < 0 No real solution; two complex imaginary solutions No x-intercepts The Discriminant and the Kinds of Solutions to ax2 + bx +c = 0

More Related