Sec 1.8 –Revisiting Quadratics Quadratic Formula with Imaginary Solutions Name:

Solve the following by completing the square. Derive the quadratic formula.

02 cxbxa

aacbbx

242

M. Winking Unit 1-8 page 22

The Quadratic Formula Solve the following by quadratic formula.

1. 019102 xx 2. 08152 xx

3. 22 8 15 0x x 4. 0673 2 xx

5. 03102 2 xx 6. 23 6 7 0x x

aacbbx

242

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The DISCRIMINANT The discriminant determines the type of solution Rational or Irrational and Real or Imaginary.

aacbbx

242

acb 42

Describe the nature of the roots using the discriminant.

1. 01482 2 xx 2. 0823 2 xx

3. 012153 2 xx 4. 07103 2 xx

5. 018248 2 xx 6. 01362 xx 8.

7. 0243412 2 xx

Positive Perfect Square

Examples = 9, 25, 49 If the discriminate is a positive perfect square then the square root can be completely eliminated when the radical is simplified and no radical will be left over. So the solution can be written as 2 REAL and RATIONAL number.

Positive Non Square

Examples = 7, 12, 22

If the discriminate is positive but NOT a perfect square then there will still be 2 REAL solutions but the solutions will be IRRATIONAL since the radical cannot be completely eliminated.

Zero

Examples = 0 If the discriminate is zero then there will be just 1 REAL RATIONAL solution because adding or subtracting 0 is equivalent.

Negative

Examples = – 3, – 9, – 12

If the discriminate is negative then there will be 2 IMAGINARY solutions (there will be an i in the solution).

acb 42 = discriminant

Give a possible value for ‘a’ such that the solution of the quadratic would have 2 real rational solutions.

0252 xax

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9. Matching (determine which graph matches which situation) Consider that 3 different quadratic equations set equal to y and graphed. Indicate what must be true about the discriminant of each original quadratic equation based on the graph. ____a. Discriminant is positive. I. II. III. ____b. Discriminant is zero.

____c. Discriminant is negative.

Applications 1. Find the value of x that would make the diagram below accurate.

2. A golf ball is hit with an initial vertical velocity of 80 fps 216 80h t t

a. How high is the ball after 2 seconds?

b. When will the ball reach 48 feet?

c. When will the ball reach 110 feet?

x+2

x x+4

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3. A baseball his hit with an initial vertical velocity of 121 fps an the ball was struck 1 foot above ground. 216 70 1h t t

a. How high is the ball after 2 seconds?

b. When will the ball reach 101 feet?

c. When will the ball reach 41 feet?

4. A person at a framing store is making a frame mat to go around a picture. The mat is a uniform 2 inches around on each side. The picture’s width is 5 less than twice the picture’s height. The entire area of the frame with picture included is 221 square inches. What are the dimensions of the picture?

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