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Algebra 1

9.8.2 Deriving the Quadratic Formula, Part 2

Algebra 19.8.2 Deriving the Quadratic Formula, Part 2

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Activity

1.

Here is one way to make sense of how the quadratic formula came about. Study the derivation until you can explain what happened in each step. Write down your explanation for each step.

Original equation

a x 2 + b x + c = 0 a x 2 + b x + c = 0

Step 1 - 4a2x2+4abx+4ac=04a2x2+4abx+4ac=0

Step 2 - 4a2x2+4abx=4ac4a2x2+4abx=4ac

Step 3 - (2ax)2+2b(2ax)=4ac(2ax)2+2b(2ax)=4ac

Step 4 - M2+2bM=4acM2+2bM=4ac

Step 5 - M2+2bM+b2=4ac+b2M2+2bM+b2=4ac+b2

Step 6 - (M+b)2=b24ac(M+b)2=b24ac

Step 7 - (M+b)2=b24ac(M+b)2=b24ac

M + b = ± b 2 4 a c M + b = ± b 2 4 a c

Step 8 - M=b±b24acM=b±b24ac

Step 9 - 2ax=b±b24ac2ax=b±b24ac

Step 10 -

x = b ± b 2 4 a c 2 a x = b ± b 2 4 a c 2 a

Are you ready for more?

Extending Your Thinking

1.

Here is another way to derive the quadratic formula by completing the square.

  • First, divide each side of the equation ax2+bx+c=0ax2+bx+c=0 by aa to get x2+bax+ca=0x2+bax+ca=0.

  • Then, complete the square for x2+bax+ca=0x2+bax+ca=0.

In problems 1 – 10, briefly explain what happens in each step by answering the question.

Original equation

x 2 + b a x + c a = 0 x 2 + b a x + c a = 0

Step 1
What constant should be subtracted from both sides?

2.

Step 2
What constant do we need to add to both sides to complete the square?

3.

Step 3
Now we can complete the square. What is the binomial that is squared on the left side?

4.

Step 4
Next, we add the fractions on the right side of the equation. What will the common denominator be?

5.

Step 5
Once the fractions on the right side of the equation are added together, what will the numerator of that new fraction be?

6.

Step 6
What is the next step necessary to isolate xx on the left side?

7.

Step 7
How can the square root be simplified?

8.

Step 8
What is the denominator of the fraction on the right side after simplification?

9.

Step 9
What is the next step I need to take to isolate xx on the left side of the equation?

10.

Step 10
What do you notice about the denominator of both fractions on the right side?

Video: Deriving the Quadratic Formula

Watch the following video to learn more about deriving the quadratic formula.

Self Check

Looking at the steps of deriving the quadratic formula, which step should follow the one below?

( x + b 2 a ) 2 = b 2 4 a c 4 a 2

  1. Subtract b 2 a from both sides.
  2. Multiply both sides by 4 a 2 .
  3. Take the square root of both sides.
  4. Complete the square.

Additional Resources

A General Way to Derive the Quadratic Formula

Here is another way to look at how to derive the quadratic formula:

We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by −1 and obtain a positive aa. Given ax2+bx+c=0ax2+bx+c=0, a0a0, we will complete the square as follows:

Step 1 - First, move the constant term to the right side of the equal sign:

ax2+bx=cax2+bx=c

Step 2 - As we want the leading coefficient to equal 1, divide through by aa:

x2+bax=cax2+bax=ca

Step 3 - Then, find 1212 of the middle term, and add (12·ba)2=b24a2(12·ba)2=b24a2 to both sides of the equal sign:

x2+bax+b24a2=b24a2cax2+bax+b24a2=b24a2ca

Step 4 - Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:

(x+b2a)2=b24ac4a2(x+b2a)2=b24ac4a2

Step 5 - Now, use the square root property, which gives:

x+b2a=±b24ac4a2x+b2a=±b24ac4a2

x+b2a=±b24ac2ax+b2a=±b24ac2a

Step 6 - Finally, add b2ab2a to both sides of the equation and combine the terms on the right side. Thus:

x=b±b24ac2ax=b±b24ac2a

Try it

Try It: A General Way to Derive the Quadratic Formula

Using the general way to derive the quadratic formula, find the step after

x+b2a=±b24ac4a2x+b2a=±b24ac4a2

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