Solutions to any quadratic equation are referred to as...

4.1 Graphical Solutions

Solutions to any quadratic equation are referred to as

_____________________.

The ____________________ of a quadratic equation are found by letting f(x) = 0 and solving for x.

These can also be found by graphing the function and determining the _______________________.

4.1 Graphical Solutions

There are three possibilities for graphical solutions.

no real x-int 1 real x-int 2 real x-int

4.1 Graphical Solutions

Use a graphing calculator to determine the x-intercepts for

y = -x2 + 8x – 16

4.1 Graphical Solutions

Use a graphing calculator to determine the x-intercepts for

y = 2 + x – x2

4.1 Graphical Solutions

Use a graphing calculator to determine the x-intercepts for

y = 2x2 + x + 2

4.1 Graphical Solutions

Use a graphing calculator to determine the x-intercepts for

A) y = 3x2 + x – 2

B) y = -x2 – 3x - 5

C) y = x2 + 8x + 16

SUBMIT ELECTRONICALLY

4.1 Graphical Solutions

The curve of a suspension bridge cable attached between the tops of two towers can be modelled by the function h(d) = 0.0025(d - 100)2 – 10, where h is the vertical distance from the top of a tower to the cable and d is the horizontal distance from the left end of the bridge, both in metres. What is the horizontal distance between the two towers? Express your answer to the nearest tenth of a metre.

4.1 Graphical Solutions

Two numbers have a sum of 9 and a product of 20. Create an equation, in the form ax2 + bx + c = 0, and determine the numbers by graphing the equation.

4.2 Factoring Quadratic Equations

To solve a quadratic equation bring everything to the same side and factor. The _____________________ are found by letting each factor equal zero and solving for the indicated variable.

Factoring from 1201:

– Remove the greatest common factor first

– For trinomials utilize the “window” method or Product-Sum

– Look for patterns such as difference of squares

4.2 Factoring Quadratic Equations

Determine the roots of the following equations:

4x2 + 8x = 5 4x2 – 2x – 12 = 0

Find the roots of 2x2 + 7x + 6 = 0 (SUBMIT ELECRONICALLY)

4.2 Factoring Quadratic Equations

Determine the roots of the following equations:

4x2 – 9 = 0 16x2 – 81y2 = 0

Find the roots of 2x2 – 18 = 0 (SUBMIT ELECRONICALLY)

4.2 Factoring Quadratic Equations

Determine the roots of the following equation:

5(x – 3)2 – 9(x – 3) – 2 = 0

4.2 Factoring Quadratic Equations

Determine the roots of the following equation:

5(x – 3)2 – 9(x – 3) – 2 = 0 *Replace (x – 3) with R

Find the roots of 3(2x + 1)2 + 4(2x + 1) – 4 = 0 (SUBMIT)

4.2 Factoring Quadratic Equations

Determine the roots of the following equation:

4(3x + 1)2 – 9(2x – 3)2 = 0 * 3x + 1 = A 2x – 3 = B

Find the roots of (3x + 5)2 – (2x – 7)2 = 0 (SUBMIT)

4.2 Factoring Quadratic Equations

Determine the roots of the following equation:

¼ x2 – x – 3 = 0

Find the roots of ½ x2 + 4x – 6 = 0 (SUBMIT)

4.2 FACTORING QUADRATIC EQUATIONS

Determine the equation if the roots are x = -2 and x = ¾

Find the equation if the roots are x = 3 and x = - ½ (SUBMIT)

4.3 Solving by Completing the Square

When a quadratic equation can not be factored the method of completing the square can be used.

When asked to determine the _________________ roots, any square roots must be written in simplest radical form. Otherwise, any radicals can be simplified using a calculator and rounded as indicated.

When taking the square root of a number, remember that ± should be used to indicate ______________________________.

4.3 Solving by Completing the Square

Find the value for c that will complete the square.

-x2 – 6x + c ½ x2 + 3x + c

-2x2 + 12x + c (SUBMIT ELECTRONICALLY)

4.3 Solving by Completing the Square

Solve (x – 1)2 = 9. Solve 9(2x + 3)2 = 1

Solve 25(x – 4)2 = 9 . (SUBMIT ELECTRONICALLY)

4.3 Solving by Completing the Square

Find the roots of -0.6x2 – 10x + 1 = 0. Round to the nearest tenth.

Find the roots of 0.6x2 + 4x – 2 = 0, to the nearest tenth. (SUBMIT)

4.3 Solving by Completing the Square

Find the exact roots of 2x2 – 10x + 5 = 0.

Find the exact roots of 2x2 + 4x – 1 = 0. (SUBMIT)

4.3 Solving by Completing the Square

Find the exact roots of ½ x2 – ¾ x + 5 = 0.

Find the exact roots of ¼ x2 + ½ x – 1 = 0. (SUBMIT)

4.3 Solving by Completing the Square

Find a quadratic equation that has roots and

Find a quadratic equation that has roots (SUBMIT)

23 23

3

4.3 Solving by Completing the Square

Solve for x by completing the square.

kx2 + 4x + 6 = 0

x2 – kx + 3 = 0 (SUBMIT ELECTRONICALLY)

4.4 The Quadratic Formula

Solve for x by completing the square.

ax2 + bx + c = 0

4.4 The Quadratic Formula

Use the quadratic formula to find the EXACT roots.

¾ x2 + x – 1 = 0

-5x2 + 4x + 2 = 0 (SUBMIT ELECTRONICALLY)

4.4 The Quadratic Formula

The _____________________ can be used to determine the nature of the roots.

no real roots 1 distinct real root 2 distinct real roots

b2 – 4ac b2 – 4ac b2 – 4ac

4.4 The Quadratic Formula

Describe the nature of the roots.

2x2 – 3x + 1 = 0 x2 + 3 = 0 3x2 + 6x = -3

Find the number of zeros for -3x2 – x + 2 = 0. (SUBMIT)

Word Problems

When dealing with applications of quadratic functions it is important to verify the validity of both roots.

Any root that does not satisfy the conditions of the problem is said to be ____________________________.

For example, dimensions can not be negative.

Another example would be _______________________________ or _______________________________.

Word Problems

You pitch a softball from a height of 1m. The ball follows a parabolic path given by the equation h(t) = - 5t2 + 15t + 50, where h is the height in metres and t is the time in seconds. When does the ball hit the ground?

h(t) = 5t2 – 30t + 45. When is the height 20m? (SUBMIT)

Word Problems

Boat A leaves the dock at 9AM and heads North at a speed of 60km/h. Three hours later Boat B leaves the same dock and heads East at a speed of 80km/h. What time is it when the boats are 500km apart?

A right triangle has one leg twice the other with a hypotenuse of 12cm. Find the lengths of the legs to the nearest mm. (SUBMIT)

Word Problems

You bought an original painting with dimensions 2ft by 3ft to hang on a wall that is 12ft by 8ft. You want the painting to cover 60% of the wall so you take it to a gallery to have it matted and framed. Find the width of the uniform mat that will surround the painting.

Frame an 8cm by 10cm picture so the area is doubled. Find the width of the frame to the nearest millimetre. (SUBMIT)