Solving Quadratic Equations by Finding Square Roots What

Solving Quadratic Equations by Finding Square Roots

What is a square root? If a number square (b 2) = another number (a), then b is the square root of a. Example: If 32 = 9, then 3 is the square root of 9

Some basics… All positive numbers have two square roots, one positive and one negative

Find the square root of numbers

Perfect Squares: Numbers whose square roots are integers or quotients of integers.

Evaluate a Radical Expression

Quadratic Equations Standard form: ax 2 + bx + c = 0 a is the leading coefficient and cannot be equal to zero. If the value of b were equal to zero, the equation becomes ax 2 + c = 0. We can solve equations is this form by taking the square root of both sides.

Key Concepts When x 2 = d If d > 0, then x 2 = d has two solutions If d = 0, then x 2 = d has one solution If d < 0, then x 2 = d has no real solution

Solving quadratics Solve each equation. a. x 2=4 b. x 2=5 c. x 2=0 d. x 2=-1 x 2=4 has two solutions, x = 2, x = -2 x 2=5 has two solutions, x =√ 5, x =- √ 5 x 2=0 has one solution, x = 0 x 2=-1 has no real solution, x = i, x = -i

Solve by rewriting equation Solve 3 x 2 – 48 = 0 3 x 2 – 48 + 48 = 0 + 48 3 x 2 = 48 3 x 2 / 3 = 48 / 3 x 2 = 16 After taking square root of both sides, x=± 4

Solving Quadratic Equations 1. Solve. 3 - 5 x 2 = -9 -3 -3 -5 x 2 = -12 -5 -5 x 2 = 2. Solve. 3(x-2)2=21 3 3 (x-2)2 = 7

More Examples! 3. Solve. 4 x 2 -6=42 +6 +6 4 x 2=48 4 4 x 2 = 12 4. Solve.

Equation of a falling object When an object is dropped, the speed with which it falls continues to increase. Ignoring air resistance, its height h can be approximated by the falling object model. h is the height in feet above the ground t is the number of seconds the object has been falling s is the initial height from which the object was dropped

Falling Objects! Use h = -16 t 2 + h 0 Height of the object after it has fallen # of seconds after the object is dropped Object’s initial height

Example The tallest building in the USA is in Chicago, Illinois. It is 1450 ft. tall. How long would it take a penny to drop from the top of the building to the ground?

Application An engineering student is in an “egg dropping contest. ” The goal is to create a container for an egg so it can be dropped from a height of 32 feet without breaking the egg. To the nearest tenth of a second, about how long will it take for the egg’s container to hit the ground? Assume there is no air resistance.

The question asks to find the time it takes for the container to hit the ground. Initial height (s) = 32 feet Height when its ground (h) = 0 feet Time it takes to hit ground (t) = unknown

Substitute 0 = -16 t 2 + 32 -32 + 0 = -16 t 2 + 32 – 32 -32 = -16 t 2 -32 / -16 = -16 t 2 / -16 2 = t 2 t = √ 2 seconds or approx. 1. 4 seconds