Welcome! Today we're exploring The Sum Formula as a Problem-Solving Equation — a tool that's going to save you a lot of time.

The sum formula $\alpha + \beta = \frac{-b}{a}$ is not just for verification — it is a powerful equation-solving tool.

When a problem tells you:

The sum of zeros, or
One zero and asks for the other

...this formula gives the answer directly — without any factoring.

In this lesson, you will see three problem types that the sum formula handles instantlyfast!:

Problem Type	What You're Given
Type 1	Sum of zeros
Type 2	One zero
Type 3	Relationship between zeros

1. Finding an unknown coefficient from a given sum

Here's a powerful insight: when a problem states the sum of zeros, setting $\frac{-b}{a}$ equal to that sum gives you a direct equation for the unknown coefficient.

No factoring needed — just algebra!

📋 Given Info

Let's see this in action.

Consider the polynomial $kx^2 - 3x + 5$ .

Here we have:

$a = k$
$b = -3$
$c = 5$

So the sum of zeros is:

\alpha + \beta = \frac{-b}{a} = \frac{-(-3)}{k} = \frac{3}{k}

✍️ Question

Question 📝

The sum of zeros of $k x^{2} - 3 x + 5$ is $1$ given.

Find $k$ .

Let's work through this step by step.

For the polynomial $k x^{2} - 3 x + 5$ , we need to identify our coefficientsYour first job with any polynomial:

$a = k$ The coefficient of x squared (coefficient of $x^{2}$ )
$b = -3$ The coefficient of x, with its sign (coefficient of $x$ )

✍️ MCQ

Choose one

In the polynomial

kx^2 - 3x + 5

, what is the value of

b

?

✓

Using the sum formulaYour essential tool for finding unknown coefficients: Sum $= \frac{-b}{a}$ Memorize this key relationship $= \frac{- (- 3)}{k} = \frac{3}{k}$ .

Notice the double negativePause and handle signs carefully here here: since $b = - 3$ , we have $-b = -(-3) = +3$ Negative of negative gives positive. So $\frac{- b}{a} = \frac{3}{k}$ .

✍️ MCQ

Choose one

When

b = -3

, what is the value of

-b

?

✓

We're told the sum equals 1. So we set up the equation:

\frac{3}{k} = 1

(Setting the known sum equal to the formula creates a solvable equation)

✍️ MCQ

Choose one

To solve

\frac{3}{k} = 1

for

k

, what operation should we perform?

✓

Solving: Multiply both sides by $k$ Standard algebra — multiply both sides to isolate the unknown:

$3 = k$

So $k = 3$ Answer!The unknown value comes out directly from the algebra.

2. Finding the other zero when one is known

Here's a useful technique:

When one zero is given and you need the other, the sum formula gives you the total.

Subtract the known zero to get the unknown one. No factoring needed!

✍️ Question

Try this 🧮

One zero of $x^2 - 4x + 1$ is $(2 + \sqrt{3})$ .

Find the other zero without factoring.key

The sum formulaYour shortcut to get the total directly gives the total sum directlyGet the total without finding individual zeros from the coefficientsUse coefficients without finding zeros first.

For $x^{2} - 4 x + 1$ : $a = 1$ , $b = - 4$ .

Sum $= \frac{- b}{a} =$ (double negative) $\frac{-(-4)}{1}$ When b is negative, negative b flips to positive $=$ $4$ This becomes your equation for solving problems.

✍️ MCQ

Choose one

If the sum of both zeros is

4

and one zero is

(2 + \sqrt{3})

, what equation would you set up to find the other zero?

✓

If one zero is $(2 + \sqrt{3})$ , the other is:

\beta = 4 - (2 + \sqrt{3})

(Subtract the known zero from the sum to find the other)

\beta = 4 - (2 + \sqrt{3}) = 4 - 2 - \sqrt{3}

(Distribute the minus sign to both terms)

✍️ MCQ

Choose one

What is

4 - 2 - \sqrt{3}

simplified?

✓

\beta = 2 - \sqrt{3}

(No quadratic formula or factoring needed)

The other zero is $(2 - \sqrt{3})$ Quick method when you know one zero and the sum — found using only the sum formulaJust subtraction from the sum!

Notice: the two zeros are $(2 + \sqrt{3})$ and $(2 - \sqrt{3})$ — conjugate surdsTwo expressions where only the sign of the surd changes.

See the pattern? One has a plusThe sign flips between the two expressions, the other has a minusThe sign flips between the two expressions. The rational part (2)The 2 remains the same in both expressions stays the same, only the sign of the surd changesPlus becomes minus or vice versa.

This is a pattern: when a polynomial has rational coefficientsThis rule only works for polynomials with rational coefficients and one surd zero, the other zero is always the conjugateYou get the second zero for free.

Rule: If $(p + \sqrt{q})$ given(The moment you calculate one surd zero) is a zero, then $(p - \sqrt{q})$ conjugateJust flip the sign without any extra work must also be a zero.

✍️ MCQ

Choose one

If a polynomial with rational coefficients has

(5 + \sqrt{2})

as one zero, what is the other zero?

✓