1. Bellringer

2. M11.D.2.1.5: Solve quadratic equations using factoring

3. Objectives
Solving by Factoring and Finding Square Roots Solving by Graphing

4. 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.

5. 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

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

8. Real World Examples
The function y = –16x 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 = –16x
0 = –16x Substitute 0 for y.
–270 = –16x2 Isolate x2.
= x2
± 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.

9. 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 = x ( x + 1 )2
36 = x2 + x2 + 2x + 1 Multiply.
0 = 2x2 + 2x – Write in standard form.

10. 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.

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