The Quadratic Formula

LAUNCH

1. Solve the following quadratic equations by hand and show all of your work.

(a) 3x²− 4x + 1 = 0 (b) 3x²+ 2x + 1 = 0

2. What method did you use to solve 1(a)? 1(b)?

3. Where did the method you used for 1(b) come from? How do you know that it gives you the correct results?

4. Could you have used the method you used for 1(b) to find the solutions for 1(a)? If so, do it and check that you arrive at the correct solution.

After having done the launch question you are well aware that this next section concerns the quadratic formula, which you are surely familiar with from your secondary school studies. We hope that you appreciate the power of this formula and that, at the same time, if you don’t know already, you are curious about where the formula comes from. You might also be wondering if we have such formulas for solving all polynomial equations. How nice that would be! While we do have formulas to find roots of cubic equations and fourth degree equations (some of which you will see later), it was proved by the mathematicians Abel and Ruffini (see Theorem 3.23) that there is no formula that will give us solutions to equations of 5th degree or higher. Some of the best minds worked on this problem but with no success. The theorem was a triumph and the solution was unexpected. It used group theory to prove the result.

In this section we concentrate on solving quadratic equations. We know from secondary school that the equation y²= a is very easy to solve: y =±√

a. Thus, if y²= 7, y =±√

7. If we can somehow reduce a quadratic equation ax²+ bx + c = 0 to the form y²= a, then solving it would be easy. The method that is often taught in secondary school is the method of completing the square. This method has applications to many different areas in mathematics other than solving quadratics.

For example, it can be used to find the center and radius of circles that are not in the “right” form.

It can be used to find key information about ellipses, parabolas, and hyperbolas (some of which find applications in astronomy). It can also be used to solve some rather complicated integrals in calculus that occur in the sciences. So we spend some time on it now.

What does it mean to complete a square? What it means is that you start with an expression of the form 1x²+ bx, and try to determine what must be added to this expression to make it the square of a binomial. What we must add is (^b₂)². That is, we add the square of half the coefficient of b. To see that this is correct, we simply check that 1x²+ bx + (₂^b)²= (x + ^b₂)², and is therefore a perfect square. Thus, if one asks what must be added to 1x²+ 5x to make it a perfect square, the answer is (⁵₂)²or ²⁵₄. Now we can verify that 1x²+ 5x +²⁵₄ is the square of (x +⁵₂)². To complete the square of y²− 6y, we add 9 (half of −6 all squared.) It is easy to check that y²− 6y + 9 is (y − 3)². Let us now illustrate a typical secondary school problem where a quadratic equation is solved by completing the square. Notice that this method requires that a, the coefficient of x², be equal to 1.

Example 3.16 Using the method of completing the square, solve the equation x²+ 6x + 1 = 0.

Solution. We subtract 1 from each side of the equation to get x²+ 6x =−1. We complete the square on the left side by adding 9. Of course, to keep the equation balanced, we need to do the same to the right hand side. Our equation becomes: x²+ 6x + 9 =−1 + 9. This is the same as (x + 3)²= 8. Thinking of (x + 3) as y, this tells us we have y²= 8 and hence y =±√

8. Replacing y by x + 3 we have, x + 3 =±√

8. So x = −3 ±√ 8.

Example 3.17 Use the method of completing the square to solve the equation 3x²+ 4x− 2 = 0

Solution. We add 2 to both sides to get

3x²+ 4x = 2.

To use the method of completing the square, we need the coefficient of x² to be 1. So we divide the equation by 3 to get

x²+4 3x = 2

3.

We add [¹₂(⁴₃)]²= ⁴₉ to both sides of the equation to get

x²+4 3x +4

9 =2 3+4

9 which just becomes

 x +2

3

2

= 10 9 .

From this we get that

It is exactly in this way that we derive the quadratic formula. Here it is for completeness.

Example 3.18 Derive the quadratic formula.

Solution. We start with the equation ax²+ bx + c = 0 where a> 0. We then subtract c from both sides to get

ax²+ bx =−c.

Since we need the coefficient of x²to be 1, we divide both sides by a to get x²+b

Now the left side of equation (3.10) is a perfect square, the square of (x +_2a^b). Thus, we have



Combining the two fractions on the right, we have



Subtracting_2a^b from both sides we get

The quadratic formula holds even if the coefficients a, b, and c, are complex numbers, but, of course, then quantities like √

b²− 4ac would lead to taking square roots of imaginary numbers.

What on earth does this mean? We will talk about this later when we discuss complex numbers in depth. Let us mention that, once we define what this means, the quadratic formula will hold for all quadratic equations, even if the coefficients are complex.

Given the pressure of completing a crowded curriculum and preparing students and preparing students for standardized exams, many teachers ponder the value of sharing this proof with their students. However, there may be students who are curious about where the quadratic formula came from and after having done several numerical examples with completing the squares this proof should not be difficult for them to follow. We offer another proof of the quadratic formula in the Student Learning Opportunities, which is much simpler. Although that proof is simpler, the method of completing the square occurs in several places in the secondary school curriculum, relating to conic sections and their transformations, which is why we addressed it here.

Student Learning Opportunities

1 Here is another way to derive the quadratic formula without getting bogged down in a lot of fractions. This might be more useful for a fraction-phobic classroom. Begin with ax²+ bx + c = 0 and multiply both sides of this equation by 4a to get 4a²x²+ 4abx + 4ac = 0. Now, subtract 4ac from both sides and add b² to both sides to get 4a²x²+ 4abx + b²= b²− 4ac.

Observe that the left side is a perfect square. Take it from there.

2 (C) For the quadratic equation ax²+ bx + c = 0, where a, b, and c are integers, the quantity b²− 4ac is called the discriminant. In secondary school the following rule is taught: If the discriminant is 0, there is only one root of the quadratic equation ax²+ bx + c = 0. If the discriminant is positive and a perfect square, then the two roots are real and rational and unequal. If the discriminant is positive and not a perfect square, the roots of the quadratic equation are irrational and unequal, and if the discriminant is negative, the roots are imagi-nary. Describe how you would justify these rules to your students. What would you tell them if they asked whether the rules were still true if b is irrational?

3 Solve the following quadratic equations by completing the square.

(a) x²− 6x = −8 (b) y²− 7y + 6 = 0

(c) z (z− 1) + 1 = 0 (d) 3z²− 2z + 1 = 0

4 (C) One of your students was asked to solve x²− 8x − 25 = 0 by completing the square.

Her work appears below. She notices that neither of her solutions work and concludes this quadratic equation has no answers. Comment on her work and on her conclusions. If she is

not correct, how would you help her to modify her work so that she gets a correct answer?

10 The length of a rectangle is 4 feet more than the width. The area is 22 square feet. Find the width.

11 (C) Using the solutions from the quadratic formula, how would you explain to your students why the sum of the roots of a quadratic equation ax²+ bx + c = 0 is−b

2a and that the product of the roots is c

a? Using this fact, how would you demonstrate how to find the sum and product of the roots of 2x²− 3x − 1 = 0? Show how you would justify that this answer was correct by finding the actual roots and adding them and multiplying them to check that the answer is correct.

12 In the previous problem you showed that the sum of the roots for the quadratic ax²+ bx + c = 0 is −b

a and that the product of the roots is c

a. We now wish to generalize this to cubic equations. Suppose that you have the cubic equation ax³+ bx²+ cx + d = 0. Dividing by a product and equate coefficients (Corollary 3.11) to conclude that r + s + t, the sum of the root is−b