Welcome! Today we're working on Translating Verbal Conditions About Zeros — a skill that turns word problems into solvable equations.

Many problems describe a relationship between zeros using words instead of numbers.

"One zero is twice the other" "The zeros differ by 3" "The product of zeros is equal to their sum"

The essential skill is translation:

What you see	What you do
Verbal condition	Convert to equation
Words about zeros	Sum or product formula

Once translated, the coefficient formulas solve the equation instantlykey benefit.

No guessing
No trial and error
Just direct calculation

1. Translating reciprocal and negative conditions

📋 Given Info

📝 Translating Verbal Conditions About Zeros

The two most common verbal conditions — reciprocal zeros and negatives of each other — each have a single-line translation.

Mastering these translations makes many problems trivial.

Key Facts:

Reciprocal zeros: product $= 1$ α·β
Negatives of each other: sum $= 0$ α+β

✍️ Question

Your Turn ✍️

For each condition, state the algebraic translation:

(a) The zeros of a quadratic are reciprocals of each other.

(b) The zeros of a quadratic are negatives of each other.

For each, state what equals what, and connect it to the coefficients of the quadratic $ax^2 + bx + c$ .

Translation dictionaryYour reference for translating verbal conditions:

Reciprocal zerosThe verbal condition you'll encounter in problems: zeros are $\alpha$ and $\frac{1}{\alpha}$ The standard form you'll use every time.

Product $= α \times \frac{1}{α} =$ $1$ keyReciprocal zeros always give this result

Coefficient equation: $\frac{c}{a} = 1$ Directly write this and solve from there

✍️ MCQ

Choose one

If the zeros of a quadratic

ax^2 + bx + c

are reciprocals of each other, which relationship must hold?

✓

Number line showing two points alpha and -alpha equidistant from origin O, with arrows indicating they are negatives of each other, sum equals zero

Negatives of each other: zeros are $\alpha$ and $-\alpha$ Mirror images across zero on the number line.

Sum $= α + (- α) =$ $0$ key!They cancel out completely

Look at the number line on the board — see how $α$ and $- α$ are mirror images across zeroNegatives of each other are mirror images?

Coefficient equation: $\frac{- b}{a} = 0$ , which means $b = 0$ answerThe algebraic condition you need

Summary: Zeros are negatives of each otherYour exam shortcut trigger $\Rightarrow$ $b = 0$ That's the algebraic condition

✍️ MCQ

Choose one

If the zeros of a quadratic

ax^2 + bx + c = 0

are negatives of each other, which coefficient must equal zero?

✓

Notice: reciprocal is about the PRODUCTReciprocals multiply to give 1, while negatives is about the SUMNegatives add up to zero. Do not confuse them.Don't mix up product and sum

2. Applying the reciprocal condition to find an unknown

The reciprocal condition becomes powerful when applied to polynomials with unknown coefficients.

When we know that product of zeros = 1 (the reciprocal condition), this translates to:

\frac{c}{a} = 1

key equation

This gives us a direct equation for the unknown.

📋 Given Info

Given Information:

Consider the polynomial $(a^2 + 9)x^2 + 13x + 6a$

We're told that one zero is the reciprocal of the other.

This means:

If the zeros are $α$ and $β$ , then $\beta = \frac{1}{\alpha}$
So their product: $\alpha \cdot \beta = 1$ key!

✍️ Question

Question:

If one zero of $(a^{2} + 9) x^{2} + 13 x + 6 a$ is the reciprocal of the other, find $a$ find.

Reciprocal zerosWhen you see this, think product equals 1: product = 1The translation you need for exams.

When one zero is the reciprocal of the otherThis verbal condition triggers the algebra, their product must equal 1.

If the zeros are $α$ and $\frac{1}{α}$ , then:

\alpha \times \frac{1}{\alpha} = 1

=1(The verbal-to-algebra translation)

This is the key translationWhat the question is really testing from the verbal condition to algebra.

✍️ MCQ

Choose one

For the polynomial

(a^2 + 9)x^2 + 13x + 6a

, if the zeros are reciprocals, what equation can we write using the product of zeros formula?

✓

For the polynomial $(a^{2} + 9) x^{2} + 13 x + 6 a$ :

Leading coefficientThe number in front of the highest x power (call it $A$ ) = $a^2 + 9$ This is our leading coefficient here
Constant termThe term with no x attached (call it $C$ ) = $6a$ Our constant term in this polynomial

✍️ MCQ

Choose one

Using the product of zeros formula

\frac{C}{A}

, what expression do we get for the product of zeros of this polynomial?

✓

Product of zeros = $\frac{C}{A}$ = $\frac{6a}{a^2 + 9}$ Using the product of zeros formula = $1$ key!Reciprocal zeros multiply to give 1

Solving for $a$ :

Cross-multiplyMultiply both sides by denominators to clear fractions: $6 a = a^{2} + 9$

RearrangeGet everything on one side, equal to zero: $a^2 - 6a + 9 = 0$ Standard form means equals zero

✍️ MCQ

Choose one

What is the value of

a^2 - 6a + 9

when

a = 3

?

✓

Recognise: this is $(a - 3)^2 = 0$ Recognize this pattern to save time

So $a = 3$ answerOur final answer.

The equation happened to be a perfect squarePerfect squares don't appear by accident in textbook problems — typical of well-designed problems. Clean answer, clean algebra.