Solving Quadratic Equations using the Quadratic Formula 9.5 Solving Quadratic Equations using the Quadratic Formula Students will be able to solve quadratic equations using the Quadratic Formula. Students will be able to interpret the discriminant. Students will be able to choose efficient methods for solving quadratic equations.

Students will be able to solve quadratic equations using the Quadratic Formula. By completing the square for the quadratic equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0, you can develop a formula that gives the solution of any quadratic equation in standard from. This formula is called the Quadratic Formula.

Students will be able to solve quadratic equations using the Quadratic Formula. The real solutions of the quadratic equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 are where 𝑎≠0 and 𝑏 2 −4𝑎𝑐≥0.

Students will be able to solve quadratic equations using the Quadratic Formula. Solve 2 𝑥 2 −5𝑥+3=0 using the Quadratic Formula. Remember 𝑎=2, 𝑏=−5, 𝑐=3 Plug into the Quadratic Formula

Students will be able to solve quadratic equations using the Quadratic Formula.

Students will be able to solve quadratic equations using the Quadratic Formula. Solve 𝑥 2 −6𝑥+5=0 using the Quadratic Formula. Remember 𝑎=1, 𝑏=−6, 𝑐=5 Plug into the Quadratic Formula

Students will be able to solve quadratic equations using the Quadratic Formula.

Students will be able to solve quadratic equations using the Quadratic Formula. You Try!! Solve −3𝑥 2 +2𝑥+7=0 using the Quadratic Formula. Remember 𝑎=−3, 𝑏=2, 𝑐=7 Plug into the Quadratic Formula

Students will be able to solve quadratic equations using the Quadratic Formula.

2. Students will be able to interpret the discriminant. The expression 𝑏 2 −4𝑎𝑐 in the Quadratic Formula is called the discriminant. Because the discriminant is under the radical symbol, you can use the value of the discriminant to determine the number of real solutions of a quadratic equation and the number of x-intercepts of the graph of the related function. 𝑏 2 −4ac>0 Two real solutions Two x-intercepts 𝑏 2 −4ac=0 One real solution One x-intercept 𝑏 2 −4ac<0 No real solutions No x-intercept

2. Students will be able to interpret the discriminant. Determine the number of real solutions of 𝑥 2 +8𝑥−3=0. 𝑏 2 −4ac= 8 2 −4∙1∙(−3) =64− −12 =64+12 =76 The discriminant is greater than 0. So, the equation has two real solutions.

2. Students will be able to interpret the discriminant. Determine the number of real solutions of 9 𝑥 2 +1=6𝑥. Write the equation in standard form: 9 𝑥 2 −6𝑥+1=0 𝑏 2 −4ac= (−6) 2 −4∙9∙1 =36−36 =0 The discriminant equals 0. So, the equation has one real solution.

2. Students will be able to interpret the discriminant. You Try!! Determine the number of real solutions of 6 𝑥 2 +2𝑥=−1. Write the equation in standard form: 6 𝑥 2 +2𝑥+1=0 𝑏 2 −4ac= (2) 2 −4∙6∙1 =4−24 =−20 The discriminant is less than 0. So, the equation has no real solutions.

Methods for Solving Quadratic Equations. 3. Students will be able to choose efficient methods for solving quadratic equations. Methods for Solving Quadratic Equations. Method Advantages Disadvantages Factoring Straight forward when the equation can be factored easily Some equations are not factorable Using Square Roots Use to solve equations of the form 𝑥 2 =𝑑 Can only be used for certain equations Completing the Square Best used when 𝑎=1 and 𝑏 is even May involve difficult calculations Quadratic Formula Can be used for any quadratic equation Gives exact solutions Takes time to do calculations