Welcome! Today we're learning The Five-Step Verification Procedure — your safety net for every quadratic problem.

Every time you find the zeros of a quadratic, you need to verify the relationship between zeros and coefficients.

This is a scored step, not optional.

But more than that, it is your safety net — if the verification fails, you know there is an error somewhere.

In this lesson, you will build a five-step procedure that prevents the two most common errors:

Error	What goes wrong
1. Non-standard form	Reading coefficients from the wrong arrangement
2. Double-negative trap	Sign errors when the constant or middle term is negative

1. Standard form before reading coefficients

The verification procedure starts with a step that many students skip — rearranging to standard form.

Skipping it causes wrong coefficient readings, which cascade into a failed verification even when the zeros are correct.

✍️ Question

Here's an expression:

6x^2 - 3 - 7x

Notice the terms are not in decreasing order of powers.

Rewrite this in standard form and identify $a$ , $b$ , $c$ with their signskey.

The non-negotiable first step: write the polynomial in standard formHighest power first, then next, then constant (decreasing powers of $x$ Highest power first, then next, then the constant).

$6 x^{2} - 3 - 7 x$ has the terms out of order.

The $- 3$ is the constantpower 0That's why it goes at the end (no $x$ )
$- 7 x$ is the $x$ term

Rearranging in decreasing powers:

6x^2 - 7x - 3

(The key pattern for standard form)

✍️ MCQ

Choose one

In the standard form

6x^2 - 7x - 3

, what is the value of

b

?

✓

Now read the coefficients:

$a = 6$ (coefficient of $x^{2}$ )
$b = -7$ The sign is part of the coefficient (coefficient of $x$ , including the negative signThe minus sign travels with the number)
$c = -3$ Same rule applies to the constant (constant term, including the negative sign)

✍️ MCQ

Choose one

If the standard form were

6x^2 + 7x - 3

, what would be the value of

b

?

✓

Write this as a separate lineTakes 5 seconds but catches errors:

a = 6, \quad b = -7, \quad c = -3

key values(This habit prevents cascading errors)

This 5-second habitQuick habit that pays off prevents errors that cascadeStops mistakes from snowballing through the entire verification.

2. The double-negative trap in -b/a

With coefficients correctly read, the next danger point is computing $\frac{- b}{a}$ when $b$ is already negative.

The formula has a minus sign, and when $b$ is also negative, you get a double negative that must be resolved explicitly.

⚠️ This is where sign errors creep in — don't skip steps!

The safe approach: Write it in two lines.

Line 1: State $b = -7$
Line 2: Compute $-b = -(-7) = +7$ key step

Then:

-\frac{b}{a} = \frac{7}{6}

✍️ Question

Your turn ✏️

For $a = 4$ , $b = - 4$ , $c = - 3$ , compute $-\frac{b}{a}$ .

Show the two-step method explicitly.

The double-negative trapWhere most students lose marks is the most common error in the chapter. Here is the foolproof method:

Line 1: Write down $b$ start here with its sign. → $b = - 4$

Line 2: Negate itWriting negative of negative 4 separately. → $-b = -(-4) = +4$ How you avoid the trap

Line 3: Divide by $a$ . → $\frac{- b}{a} = \frac{4}{4} =$ $1$ answer

✍️ MCQ

Choose one

If

b = -4

, what is

-b

?

✓

Common Mistake: Trying to compute $\frac{- (- 4)}{4}$ in one step

When you rush, you might write $\frac{-4}{4} = -1$ Double negative becomes positive — forgetting that negating a negative gives a positiveNegating a negative gives a positive!

Why the Two-Step MethodForces the sign change to be visible Works:

Line 1: $b = -4$ Write it as a separate line

Line 2: $-b = -(-4) = +4$ Sign change becomes visible on paper

Now: $\frac{- b}{a} = \frac{+ 4}{4} = 1$

The two-step method makes the sign change visiblekey!You cannot miss the double negative and prevents the error.

✍️ MCQ

Choose one

A student computed

-\frac{b}{a}

for

b = -4

and

a = 4

and got

-1

. What mistake did they make?

✓

This applies whenever $b$ is negativeWhen b is negative, that's when you need to be careful. When $b$ is positiveWhen b is positive, there's no trap at all (like $b = 5$ ), there's no double negative to worry about: $\frac{- b}{a} = \frac{- 5}{a}$ . StraightforwardIt's straightforward when b is positive.

✍️ MCQ

Choose one

When do you NEED to use the two-step method for computing

-\frac{b}{a}

?

✓

3. Complete verification with surd zeros

Great progress! 🎯

Standard form and sign handling are locked in.

Now we do a complete verification from start to finish, including a case with surd zeros where the sum cancels to a simple number.

📋 Given Info

Here's what you're working with:

For $p (x) = x^{2} - 2$ with zeros $\sqrt{2}$ and $- \sqrt{2}$ :

Standard form: $x^{2} + 0 x - 2$
Coefficients:
- $a = 1$
- $b = 0$
- $c = -2$

✍️ Question

Your Task ✍️

Verify the zeros-coefficients relationship for $x^2 - 2$ with zeros $\sqrt{2}$ and $-\sqrt{2}$ .

Show all five steps5:

Write in standard form
Identify $a$ , $b$ , $c$
Compute sum and product from zeros
Compute $- b / a$ and $c / a$ from formulas
Compare and verify

Here is the complete five-step procedureYour systematic checklist for checking zeros applied to $x^{2} - 2$ with zeros $\sqrt{2}$ and $- \sqrt{2}$ .

Step 1: Standard form.Writing in decreasing powers of x

$x^{2} - 2$

The $x$ term is missinghidden!Missing terms need coefficient zero, so we write it explicitly:

$x^{2} + 0 x - 2$

✍️ MCQ

Choose one

From the standard form

x^2 + 0x - 2

, what are the values of

a

,

b

, and

c

?

✓

Step 2: Read coefficients. From $x^{2} + 0 x - 2$ :

$a = 1$ Match a to the x squared term (coefficient of $x^{2}$ )
$b = 0$ (Match b to the x term) (coefficient of $x$ )
$c = -2$ The sign travels with the number (constant term)

The missing $x$ termMissing doesn't mean absent means $b = 0$ keyWrite b equals zero and continue. This is important — when a term is "missing," its coefficient is zero, not absent.

✍️ MCQ

Choose one

When

b = 0

, what is the value of

-\frac{b}{a}

?

✓

Step 3: Compute from zeros.

Sum $= \sqrt{2} + (- \sqrt{2}) =$ $0$ Negative pairs always cancel out. (The surds cancel.Watch for this pattern in exams)

Product $= \sqrt{2} \times (- \sqrt{2}) = - (\sqrt{2})^{2} =$ $-2$ Surd times its negative gives negative of the square.

✍️ MCQ

Choose one

We have

a = 1

,

b = 0

,

c = -2

. What is

-\frac{b}{a}

?

✓

Step 4This is the verification step: Compute from formulas.Use formulas, then compare to actual zeros

$-\frac{b}{a}$ The sum formula from coefficients $= - \frac{0}{1} =$ $0$ match!First confirmation — it matches!

$\frac{c}{a}$ (The product formula from coefficients) $= \frac{- 2}{1} =$ $-2$ match!Second match — verification complete!

✍️ MCQ

Choose one

In Step 4, we computed

-\frac{b}{a}

and

\frac{c}{a}

. What is the purpose of Step 5 in the verification procedure?

✓

Step 5: CompareWhere you check if values match.

Sum: $0 = 0$ . ✓ Verified.✓Calculated matches formula

Product: $- 2 = - 2$ . ✓ Verified.✓Calculated matches formula

Both relationships hold — the verification is complete!

Key observation: $b = 0$ corresponds to a sum of $0$ . When zeros are negatives of each other (like $\sqrt{2}$ and $-\sqrt{2}$ Negatives of each other cancel when added), their sum cancels out, and the $x$ -coefficient becomes $0$ No x term in the polynomial.

This is why the polynomial $x^{2} - 2$ has no $x$ term — the zeros are symmetric about zero!Zeros are symmetric about the origin

✍️ MCQ

Choose one

If a quadratic has zeros

5

and

-5

, what is the value of

b

(the coefficient of

x

)?

✓

4. Recognising patterns: what zero coefficients tell you

Through verification, we start noticing patterns:

Coefficient	What it means
$b = 0$	The zeros sum to 0 (they're negatives of each other)
$c = 0$	One zero is 0key insight itself

These structural insights connect the algebra to the geometry and help you check answers instantly.

📋 Given Info

For $p (y) = 5 y^{2} + 10 y$ , the standard form is:

$5 y^{2} + 10 y + 0$

Coefficients:

$a = 5$
$b = 10$
$c = 0$

Zeros: $0$ and $-2$ given

✍️ Question

Verify the zeros-coefficients relationship for $5y^2 + 10y$ with zeros $0$ and $- 2$ .

What does $c = 0$ tell you about the zeros?

Let's verify $5y^2 + 10y$ with zeros $0$ and $-2$ .

Step 1 — Standard form: $5 y^{2} + 10 y + 0$

Notice the constant term is missingWhen there's no constant term, c equals 0 — this means $c = 0$ You must write it explicitly as plus 0.

Step 2 — Identifying the coefficients:

$a = 5$ Always identify all three, even when one is zero (coefficient of $y^{2}$ )
$b = 10$ Always identify all three, even when one is zero (coefficient of $y$ )
$c = 0$ c is zero but still must be identified (constant term)

✍️ MCQ

Choose one

One of the zeros is

0

. When you multiply any number by

0

, what do you get?

✓

Step 3: Calculate sum and product from zeros

From zeros: Sum $= 0 + (-2) = -2$ . Product $= 0 \times (-2) = 0$ .

Step 4: Calculate using coefficient formulas

From formulas: $-\frac{b}{a} = -\frac{10}{5} = -2$ Formula for sum of zeros. $\frac{c}{a} = \frac{0}{5} = 0$ Formula for product of zeros.

✍️ MCQ

Choose one

Before we compare, what should the sum from zeros (

-2

) equal if our verification is correct?

✓

Step 5: Compare and verify

Sum: $-2 = -2$ . ✓ Verified. Product: $0 = 0$ . ✓ Verified.

Key Insight: When $c = 0$ When there's no constant term, the product of zeros is $\frac{c}{a} = 0$ , which means at least one zero must be 0One of the zeros has to be zero. This is why $y = 0$ Zero is immediately a root is a zero of $5y^2 + 10y$ No constant term means zero is a root.

✍️ Yes/No

Yes or No?

If a quadratic has

c = 0

, can both zeros be non-zero?

✓

Let's annotate the graph to see how the zeros we found connect directly to the coefficient formulas. When you see the sum and product written right next to the zeros, the verification becomes visual.

✍️ MCQ

Choose one

The product of zeros is

0

. What does this tell us about at least one of the zeros?

✓

The pattern: $c = 0$ When c is zero, we know something special means product of zeros $= 0$ The product of the zeros equals zero. Since a product is $0$ only when at least one factor is $0$ A product can only be zero if one factor is zero, this tells us $x = 0$ is a zeroSo x equals 0 must be a zero of the polynomial.

In fact, whenever you see no constant termSpotting no constant term is the key (like $5 y^{2} + 10 y$ ), you can immediately say " $y = 0$ is a zeroYou can instantly say one zero is zero" by factoring out $y$ Just factor out the variable: $5 y (y + 2)$ .

✍️ MCQ

Choose one

If a quadratic has the form

ax^2 + bx

(no constant term), which of the following is ALWAYS true?

✓

Similarly, $b = 0$ When b is zero, something interesting happens means sum of zeros $= 0$ The sum of the zeros equals zero, which happens when the zeros are negatives of each otherThe zeros are negatives of each other (like $\sqrt{2}$ and $-\sqrt{2}$ Root 2 and negative root 2 cancel out perfectly).