8.1 – Solving Quadratic Equations Many quadratic equations can not be solved by factoring. Other techniques are required to solve them. Examples: x2 = 20 5x2 + 55 = 0 ( x + 2)2 = 18 ( 3x – 1)2 = –4 x2 + 8x = 1 2x2 – 2x + 7 = 0

8.1 – Solving Quadratic Equations Square Root Property If b is a real number and if a2 = b, then a = ±√¯‾. b x2 = 20 5x2 + 55 = 0 x = ±√‾‾ 20 5x2 = –55 x = ±√‾‾‾‾ 4·5 x2 = –11 x = ± 2√‾ 5 x = ±√‾‾‾ –11 x = ± i√‾‾‾ 11

8.1 – Solving Quadratic Equations Square Root Property If b is a real number and if a2 = b, then a = ±√¯‾. b ( x + 2)2 = 18 ( 3x – 1)2 = –4 x + 2 = ±√‾‾ 18 3x – 1 = ±√‾‾ –4 x + 2 = ±√‾‾‾‾ 9·2 3x – 1 = ± 2i x +2 = ± 3√‾ 2 3x = 1 ± 2i x = –2 ± 3√‾ 2

8.1 – Solving Quadratic Equations Completing the Square Review: ( x + 3)2 x2 – 14x x2 + 2(3x) + 9 x2 + 6x + 9 x2 – 14x + 49 x2 + 6x ( x – 7) ( x – 7) ( x – 7)2 x2 + 6x + 9 ( x + 3) ( x + 3) ( x + 3)2

8.1 – Solving Quadratic Equations Completing the Square x2 + 9x x2 – 5x

8.1 – Solving Quadratic Equations Completing the Square x2 + 8x = 1 x2 + 8x = 1

8.1 – Solving Quadratic Equations Completing the Square 5x2 – 10x + 2 = 0 5x2 – 10x = –2 or

8.1 – Solving Quadratic Equations Completing the Square 2x2 – 2x + 7 = 0 2x2 – 2x = –7 or

8.2 – Solving Quadratic Equations The Quadratic Formula The quadratic formula is used to solve any quadratic equation. Standard form of a quadratic equation is: The quadratic formula is:

8.2 – Solving Quadratic Equations The Quadratic Formula

8.2 – Solving Quadratic Equations The Quadratic Formula

8.2 – Solving Quadratic Equations The Quadratic Formula The quadratic formula is used to solve any quadratic equation. Standard form of a quadratic equation is: The quadratic formula is:

8.2 – Solving Quadratic Equations The Quadratic Formula

8.2 – Solving Quadratic Equations The Quadratic Formula

8.2 – Solving Quadratic Equations The Quadratic Formula

8.2 – Solving Quadratic Equations The Quadratic Formula

8.2 – Solving Quadratic Equations The Quadratic Formula and the Discriminate The discriminate is the radicand portion of the quadratic formula (b2 – 4ac). It is used to discriminate among the possible number and type of solutions a quadratic equation will have. b2 – 4ac Name and Type of Solution Positive Two real solutions Zero One real solutions Negative Two complex, non-real solutions

8.2 – Solving Quadratic Equations The Quadratic Formula and the Discriminate b2 – 4ac Name and Type of Solution Positive Two real solutions Zero One real solutions Negative Two complex, non-real solutions Positive Two real solutions

8.2 – Solving Quadratic Equations The Quadratic Formula and the Discriminate b2 – 4ac Name and Type of Solution Positive Two real solutions Zero One real solutions Negative Two complex, non-real solutions Negative Two complex, non-real solutions

8.2 – Solving Quadratic Equations The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. x + 2 x 20 feet

8.2 – Solving Quadratic Equations The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. x + 2 (x + 2)2 + x2 = 202 x2 + 4x + 4 + x2 = 400 x 20 feet 2x2 + 4x + 4 = 400 2x2 + 4x – 369 = 0 The Pythagorean Theorem 2(x2 + 2x – 198) = 0 a2 + b2 = c2

8.2 – Solving Quadratic Equations The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 2(x2 + 2x – 198) = 0 x + 2 x 20 feet The Pythagorean Theorem a2 + b2 = c2

8.2 – Solving Quadratic Equations The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. x + 2 x 20 feet The Pythagorean Theorem a2 + b2 = c2

8.2 – Solving Quadratic Equations The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. x + 2 x 20 feet 28 – 20 = 8 ft The Pythagorean Theorem a2 + b2 = c2