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Presentation transcript:

Bellringer

M11.D.2.1.5: Solve quadratic equations using factoring

Objectives Solving by Factoring and Finding Square Roots Solving by Graphing

Vocabulary The standard form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0. If ab = 0, then a = 0 or b = 0. ex. If (x + 3)(x – 7) = 0, then (x + 3) = 0 or (x – 7) = 0.

Solving by Factoring 3x2 – 20x – 7 = 0 3x2 – 20x – 7 = 0 Write in standard form. 3x2 – 21x + x – 7 = 0 Rewrite the bx term. 3x(x – 7) + (x – 7) = 0 Find common factors. (3x + 1)(x – 7) = 0 Factor using the Distributive Property. 3x + 1 = 0 or x – 7 = 0 Use the Zero-Product Property. 1 3 x = – or x = 7 Solve for x. The solutions are – and 7. 1 3

Continued (continued) Check: 3x2 – 20x = 7 3x2 – 20x = 7 3 – 2 – 20 – 7 3(7)2 – 20(7) 7 1 3 1 3 + 7 147 – 140 7 1 3 20 3 7 = 7 7 = 7

Solving by Finding Square Roots Solve 6x2 – 486 = 0. 6x2 – 486 = 0 Rewrite in the form ax2 = c. = Isolate x2. 6x2 6 486 x2 = 81 Simplify. x = ±9 Take the square root of each side.

Real World Examples The function y = –16x2 + 270 models the height y in feet of a heavy object x seconds after it is dropped from the top of a building that is 270 feet tall. How long does it take the object to hit the ground? y = –16x2 + 270 0 = –16x2 + 270 Substitute 0 for y. –270 = –16x2 Isolate x2. 16.875 = x2 ±4.1 x Take the square root of each side. The object takes about 4.1 seconds to hit the ground. Check: Is the answer reasonable? The negative number –4.1 is also a solution to the equation. However, since a negative value for time has no meaning in this case, only the positive solution is reasonable.

Solving by Graphing A carpenter wants to cut a piece of plywood in the shape of a right triangle. He wants the hypotenuse of the triangle to be 6 feet long, as shown in the diagram. About how long should the perpendicular sides be? Relate: From the Pythagorean Theorem we know for a right triangle that the hypotenuse squared equals the sum of the squares of the other two sides. Define: Let x = the shorter leg. Then x + 1 = the longer leg. Write: 62 = x2 + ( x + 1 )2 36 = x2 + x2 + 2x + 1 Multiply. 0 = 2x2 + 2x – 35 Write in standard form.

Continued (continued) Graph the related function y = 2x2 + 2x – 35. Use the CALC feature to find the positive solution. The sides of the triangle are 3.7 ft and 4.7 ft.

Homework 5-5 Pg 270 # 1, 2, 7, 8,15, 23, 24