Deriving the Quadratic Formula, Part 1

9.8.1 Deriving the Quadratic Formula, Part 1

9.8.1 • Deriving the Quadratic Formula, Part 1

Activity

One way to solve the quadratic equation $x^{2} + 5 x + 3 = 0$ is by completing the square. A partially solved equation is shown here. Study Steps 1 – 4.

Original equation $x^{2} + 5 x + 3 = 0$

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 $x^{2}$ term. In this case, that is just 4 times 1, which is 4.)

$4 x^{2} + 20 x + 12 = 0$

Step 2 - Subtract 12 from each side.

$4 x^{2} + 20 x = - 12$

Step 3 - Rewrite

$4 x^{2}$ as $(2 x)^{2}$ and $20 x$ as $10 (2 x)$ .

$(2 x)^{2} + 10 (2 x) = - 12$

Step 4 - Use $P$ as a placeholder for $2 x$ .

$P^{2} + 10 P = - 12$

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

Step 5 - $_______________________$

$P^{2} + 10 P +____^{2} = - 12 +____^{2}$

Step 6 - $_______________________$

$(P +____)^{2} = - 12 +____^{2}$

Step 7 - $_______________________$

$\sqrt{(P +____)^{2}} = \sqrt{- 12 +____^{2}}$

$P +____= \pm \sqrt{- 12 +____^{2}}$

Step 8 - $_______________________$

$P =____\pm \sqrt{- 12 +____^{2}}$

Step 9 - $_______________________$

$P =____\pm \sqrt{____^{2} - 12}$

Step 10 -

$_______________________$

$2 x =____\pm \sqrt{____^{2} - 12}$

Step 11

$_______________________$

$x =$

Compare your answer:

Step 5 -

Add to complete the square.

$P^{2} + 10 P + 5^{2} = - 12 + 5^{2}$

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

$(P + 5)^{2} = - 12 + 5^{2}$

Step 7 - Find the square root of both sides of the equation.

$\sqrt{(P + 5)^{2}} = \sqrt{- 12 + 5^{2}}$

$P + 5 = \pm \sqrt{- 12 + 5^{2}}$

Step 8 - Subtract 5 from both sides to isolate.

$P = - 5 \pm \sqrt{- 12 + 5^{2}}$

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

$P = - 5 \pm \sqrt{5^{2} - 12}$

Step 10 -

Re-substitute $2 x$ for $P$ .

$2 x = - 5 \pm \sqrt{5^{2} - 12}$

Step 11

Divide each side by 2 to isolate $x$ .

$x = \frac{- 5 \pm \sqrt{5^{2} - 12}}{2}$

Explain how the solution is related to the quadratic formula.

Compare your answer:

The numbers in the solution are the values of $a$ , $b$ , and $c$ in the original equation, $x^{2} + 5 x + 3 = 0$ , which is in the form of $a x^{2} + b x + c = 0$ . The -5 is $- b$ . The $5^{2}$ is $b^{2}$ . The 12, which is $4 (3) (1)$ , is the value of $4 a c$ . The $a$ in the original equation is 1, and $2 (1)$ is 2, which is the number in the denominator in the solution.

Self Check

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

$x^2+8x+16= -11+16$
$x^2+8x+16=5$

What should he do next?

Complete the square again.
Factor the perfect square trinomial.
Take the square root of both sides.
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 $x^{2} + 7 x + 4 = 0$

$4 x^{2} + 28 x + 16 = 0$

Step 2 - Subtract 16 from each side.

$4 x^{2} + 28 x = - 16$

Step 3 - Rewrite.

$4 x^{2}$ as $(2 x)^{2}$ and $28 x$ as $14 (2 x)$

$(2 x)^{2} + 14 (2 x) = - 16$

Step 4 - Use $P$ as a placeholder for $2 x$ .

$P^{2} + 14 P = - 16$

Step 5 - Add to complete the square.

$P^{2} + 14 P + 7^{2} = - 16 + 7^{2}$

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

$(P + 7)^{2} = - 16 + 7^{2}$

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

$P + 7 = \pm \sqrt{- 16 + 7^{2}}$

Step 8 - Subtract 7 from both sides to isolate.

$P = - 7 \pm \sqrt{- 16 + 7^{2}}$

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

$P = - 7 \pm \sqrt{7^{2} - 16}$

Step 10 - Re-substitute $2 x$ for $P$ .

$2 x = - 7 \pm \sqrt{7^{2} - 16}$

Step 11 - Divide each side by 2 to isolate $x$ .

$x = \frac{- 7 \pm \sqrt{7^{2} - 16}}{2}$

Notice the original equation, $y = x^{2} + 7 x + 4$ , has $a = 1$ , $b = 7$ , $c = 4$ , and the last step is of the form $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , which is the quadratic formula.

Try it

Deriving the Quadratic Formula Using an Example

Jaiden was deriving the quadratic formula from the quadratic $x^{2} + 11 x + 5 = 0$ . What is the step after the step shown below?

$(x + \frac{11}{2})^{2} = \frac{11^{2} - 4 (1) (5)}{4 (1)}$

$(x + \frac{11}{2})^{2} = \frac{121 - 20}{4}$

Here is how to find the next step:

To isolate $x$ , take the square root of both sides:

$\sqrt{(x + \frac{11}{2})^{2}} = \sqrt{\frac{121 - 20}{4}}$