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Algebra 1

9.8.1 Deriving the Quadratic Formula, Part 1

Algebra 19.8.1 Deriving the Quadratic Formula, Part 1

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Activity

1.

One way to solve the quadratic equation x2+5x+3=0x2+5x+3=0 is by completing the square. A partially solved equation is shown here. Study Steps 1 – 4.

Original equation x2+5x+3=0x2+5x+3=0

Step 1 - Multiply each side by 4. This helps us avoid some tricky fractions. (In general, multiply each side of the equation by 4 times the coefficient of the x2x2 term. In this case, that is just 4 times 1, which is 4.)

4 x 2 + 20 x + 12 = 0 4 x 2 + 20 x + 12 = 0

Step 2 - Subtract 12 from each side.

4 x 2 + 20 x = 12 4 x 2 + 20 x = 12

Step 3 - Rewrite

4x24x2 as (2x)2(2x)2 and 20x20x as 10(2x)10(2x).

( 2 x ) 2 + 10 ( 2 x ) = 12 ( 2 x ) 2 + 10 ( 2 x ) = 12

Step 4 - Use PP as a placeholder for 2x2x.

P 2 + 10 P = 12 P 2 + 10 P = 12

For steps 5 – 11, let PP serve as a placeholder for 2x2x, continue to solve for xx but without evaluating any part of the expression. For each step, explain the mathematical operation(s) being performed.

Step 5 - ______________________________________________

P 2 + 10 P + _ _ _ _ 2 = 12 + _ _ _ _ 2 P 2 + 10 P + _ _ _ _ 2 = 12 + _ _ _ _ 2

Step 6 - ______________________________________________

( P + _ _ _ _ ) 2 = 12 + _ _ _ _ 2 ( P + _ _ _ _ ) 2 = 12 + _ _ _ _ 2

Step 7 - ______________________________________________

( P + _ _ _ _ ) 2 = 12 + _ _ _ _ 2 ( P + _ _ _ _ ) 2 = 12 + _ _ _ _ 2

P + _ _ _ _ = ± 12 + _ _ _ _ 2 P + _ _ _ _ = ± 12 + _ _ _ _ 2

Step 8 - ______________________________________________

P = _ _ _ _ ± 12 + _ _ _ _ 2 P = _ _ _ _ ± 12 + _ _ _ _ 2

Step 9 - ______________________________________________

P = _ _ _ _ ± _ _ _ _ 2 12 P = _ _ _ _ ± _ _ _ _ 2 12

Step 10 -

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

2 x = _ _ _ _ ± _ _ _ _ 2 12 2 x = _ _ _ _ ± _ _ _ _ 2 12

Step 11

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

x = x =

2.

Explain how the solution is related to the quadratic formula.

Self Check

After completing the square, Odem used the following steps when deriving the quadratic formula.

x 2 + 8 x + 16 = 11 + 16
x 2 + 8 x + 16 = 5

What should he do next?

  1. Complete the square again.
  2. Factor the perfect square trinomial.
  3. Take the square root of both sides.
  4. Subtract 16 from both sides.

Additional Resources

Deriving the Quadratic Formula Using an Example

Here is another example that goes through the process of deriving the quadratic formula:

Original equation x2+7x+4=0x2+7x+4=0

Step 1 - Multiply each side by 4. This helps us avoid some tricky fractions. (In general, multiply each side of the equation by 4 times the coefficient of the x2x2 term. In this case, that is just 4 times 1, which is 4.)

4x2+28x+16=04x2+28x+16=0

Step 2 - Subtract 16 from each side.

4x2+28x=164x2+28x=16

Step 3 - Rewrite.

4x24x2 as (2x)2(2x)2 and 28x28x as 14(2x)14(2x)

(2x)2+14(2x)=16(2x)2+14(2x)=16

Step 4 - Use PP as a placeholder for 2x2x.

P2+14P=16P2+14P=16

Step 5 - Add to complete the square.

P2+14P+72=16+72P2+14P+72=16+72

Step 6 - Write the left side as a squared factor.

(P+7)2=16+72(P+7)2=16+72

Step 7 - Find the square root of the expression on the right.

P+7=±16+72P+7=±16+72

Step 8 - Subtract 7 from both sides to isolate.

P=7±16+72P=7±16+72

Step 9 - Rearrange the expression under the square root sign.

P=7±7216P=7±7216

Step 10 - Re-substitute 2x2x for PP.

2x=7±72162x=7±7216

Step 11 - Divide each side by 2 to isolate xx.

x=7±72162x=7±72162

Notice the original equation, y=x2+7x+4y=x2+7x+4, has a=1a=1, b=7b=7, c=4c=4, and the last step is of the form x=b±b24ac2ax=b±b24ac2a, which is the quadratic formula.

Try it

Try It: Deriving the Quadratic Formula Using an Example

Jaiden was deriving the quadratic formula from the quadratic x2+11x+5=0x2+11x+5=0. What is the step after the step shown below?

(x+112)2=1124(1)(5)4(1)(x+112)2=1124(1)(5)4(1)

(x+112)2=121204(x+112)2=121204

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