1 Topic 7.3.1 The Quadratic Formula. 2 Topic 7.3.1 The Quadratic Formula California Standards: 19.0 Students know the quadratic formula and are familiar

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Topic 7.3.1Topic 7.3.1

The Quadratic FormulaThe Quadratic Formula

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Topic7.3.1


California Standards:19.0 Students know the quadratic formula and are familiar with its proof by completing the square.

20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.

What it means for you:You’ll use the quadratic formula to solve quadratic equations — and you’ll derive the quadratic formula itself.

Key words:• quadratic formula• completing the square

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Topic7.3.1


You can also use the quadratic formula to solve quadratic equations. It works every time.

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Topic7.3.1

Quadratic Equations can be in Any Variable


The standard form for a quadratic equation is:

ax2 + bx + c = 0 (a 0)

Any quadratic equation can be written in this form by, if necessary, rearranging it so that zero is on one side.

A lot of the quadratic equations you will see may contain a variable other than x — but they are still quadratic equations like the one above, and can be solved in the same way.

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Topic7.3.1

Example 1

Solution follows…


Solution continues…

a) Here, a = 1, b = –4, and c = 3.

So using the zero property:(x – 3) = 0 or (x – 1) = 0, or x = 3 or x = 1

This equation factors to give (x – 3)(x – 1) = 0

Solution

Find the solutions of the following quadratic equations:a) x2 – 4x + 3 = 0 b) y2 – 4y + 3 = 0

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Topic7.3.1

Example 1


Solution (continued)

Find the solutions of the following quadratic equations:a) x2 – 4x + 3 = 0 b) y2 – 4y + 3 = 0

b) Here, the variable is y rather than x — but that does not affect the solutions.

Again a = 1, b = –4, and c = 3, so it is the same equation as in a), and will have the same solutions: y = 3 or y = 1

You can see that the two quadratic equations are really the same — only the variables have changed.

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Topic7.3.1

The Quadratic Formula Solves Any Quadratic Equation


The solutions to the quadratic equation ax2 + bx + c = 0 are given by the quadratic formula:

You can derive the formula by completing the square.

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Topic7.3.1


ax2 + bx + c = 0 ax2 + bx = –c

ax2 + x = –

3. Divide the equation by a

1. Start with the standard form 2. Subtract c

4. Complete the square

5. Add the fractions

7. Include positive and negative roots

6. Simplify

8. Rearrange to give the quadratic formula

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Topic7.3.1

Example 2

Solution follows…


Solution


Solve x2 – 5x – 14 = 0 using the quadratic formula.

Start by writing down the values of a, b, and c: a = 1, b = –5, and c = –14

Now very carefully substitute these into the quadratic formula.

Then simplify to find the values of x.

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Topic7.3.1

Example 2



Solve x2 – 5x – 14 = 0 using the quadratic formula.

So or

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Topic7.3.1

Example 3

Solution follows…


Solution


Solve 2x2 – 3x – 2 = 0 using the quadratic formula.

Start by writing down the values of a, b, and c: a = 2, b = –3, and c = –2



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Topic7.3.1

Example 2



Solve 2x2 – 3x – 2 = 0 using the quadratic formula.

So or

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Topic7.3.1

Example 4

Solution follows…


Solution


Solve 2x2 – 11x + 13 = 0 using the quadratic formula.

Start by writing down the values of a, b, and c: a = 2, b = –11, and c = 13



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Topic7.3.1

Example 4



Solve 2x2 – 11x + 13 = 0 using the quadratic formula.

So or

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Topic7.3.1

Guided Practice

Solution follows…


Use the quadratic formula to solve each of the following equations.

1. x2 – 2x – 143 = 0

2. 2x2 + 3x – 1 = 0

3. x2 + 2x – 1 = 0

4. x2 + 3x + 1 = 0

5. 2x2 – 5x + 2 = 0

x = 13, –11

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Topic7.3.1

Guided Practice

Solution follows…



6. 3x2 – 2x – 3 = 0

7. 2x2 – 7x – 3 = 0

8. 6x2 – x – 1 = 0

9. 18x2 + 3x – 1 = 0

10. 4x2 – 5x + 1 = 0

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Topic7.3.1

Guided Practice

Solution follows…


11. The equation 2x2 – 7x – 4 = 0 factors to (2x + 1)(x – 4) = 0.

Using the zero product property we can find that x = – or x = 4. Verify this using the quadratic formula.

12. The height of a triangle is 4 ft more than 4 times its base

length. If the triangle’s area is ft2, find the length of its base.

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Topic7.3.1

Independent Practice

Solution follows…



1. 5x2 – 11x + 2 = 0

2. 2x2 + 7x + 3 = 0

3. 7x2 + 6x – 1 = 0

4. x2 – 7x + 5 = 0

5. 10x2 + 7x + 1 = 0

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Topic7.3.1


Solution follows…



6. 3y2 – 8y – 3 = 0

7. 5x2 – 2x – 3 = 0

8. 4x2 + 3x – 5 = 0

9. 4t2 + 7t – 2 = 0

10. 6m2 + m – 1 = 0

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Topic7.3.1


Solution follows…



11. 2x2 – x = 1

12. 3x2 – 5x = 2

13. 2x2 + 7x = 4

14. 4x2 + 17x = 15

15. 4x2 – 13x + 3 = 0

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Topic7.3.1


Solution follows…



16. 4x2 – 1 = 0

17. 25x2 – 9 = 0

18. 4x2 + 15x = 4

19. 10x2 + 1 = 7x

20. 16x2 + 3 = 26x

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Topic7.3.1


Solution follows…


Solve these equations by factoring and using the zero product property, then verify the solutions by solving them with the quadratic formula.

21. x2 + 4x + 4 = 0 22. 4y2 – 9 = 0

23. x2 – x – 12 = 0 24. 2x2 – 3x – 9 = 0

25. 6x2 + 29x = 5 26. 7x2 + 41x = 6

x = –2

x = 4, –3

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Topic7.3.1


Solution follows…


27. The length of a rectangle is 20 cm more than 4 times its width. If the rectangle has an area of 75 cm2, find its dimensions.

28. The equation h = –14t2 + 12t + 2 gives the height of a tennis ball t seconds after being hit. How long will the ball take before it hits the ground?

2.5 cm by 30 cm

1 second

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Topic7.3.1


Round UpRound Up

The quadratic formula looks quite complicated, but don’t let that put you off.

If you work through the derivation of the formula then you should see exactly why it contains all the elements it does.

Documents

1 Topic 7.3.1 The Quadratic Formula. 2 Topic 7.3.1 The Quadratic Formula California Standards: 19.0 Students know the quadratic formula and are familiar