After simplifying x(x+1)=x^2+2, is the equation quadratic?

No. The squared terms cancel, leaving x-2=0.

Is 2 a root of x^2-x=0?

No. Substitution gives 2=0, which is false.

Which root comes from x minus 2?

2. Subtracting 2 from 2 makes that factor zero.

Which root can describe this rectangle's width?

2 cm. The other root is negative, so it cannot measure this side.

How many real roots does this equation have?

None. The formula requires a square root of a negative number.

Besides zero, what discriminant values allow real roots?

Positive values. Their square roots are real, giving two different roots.

Teach Quadratic Equations in six steps.

For each idea: show it, ask before you tell, then say one line.

Gather scrap paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

A garden's area and a rule linking length to width can determine its dimensions.
A frame's border changes both dimensions, so its area involves a quadratic.
A thrown ball's height can lead to different times when it reaches that height.

What to teach

Recognise and solve

Quadratic equations in standard formAfter simplifying, x has whole-number powers only, with two as the highest.Roots of a quadratic equationValues that make the equation true when substituted.Solution of a quadratic equation by factorisationWhen a product is zero, set each factor to zero and solve.Problems involving quadratic equationsTurn a situation into an equation, then choose answers that fit.

Predict and choose the roots

Nature of rootsThe discriminant distinguishes different, equal or absent real roots.Condition for equal rootsChoose a coefficient that makes the discriminant zero.

How to teach it

Recognise and solve

$x(x+1)=2$

$x^2+x=2$

1. Quadratic equations in standard form

After simplifying, x has whole-number powers only, with two as the highest.

Show: Write these lines. Say, “x names the unknown; x squared means x times x.”
Ask: What equation follows when we subtract 2 from both sides?
Say: Subtracting 2 gives $x^2+x-2=0$ , with a nonzero squared term.
Used in: Frame makers use area equations to find uniform border widths.

If your child says…“Any equation showing a squared term is quadratic.”

Ask: If the squared terms cancel completely, what squared term remains?
What’s true: A squared term must survive after simplifying, with no higher powers remaining.

$x^2-x=0$

$1^2-1=0$

$1-1=0$

2. Roots of a quadratic equation

Values that make the equation true when substituted.

Show: Write these lines. Say, “Replace every x with 1, including the x being squared.”
Ask: What equality comes next, and is it true?
Say: Both sides become zero: $0=0$ , so 1 is a root.
Used in: Checking a calculated travel speed against the travel time catches unsuitable answers.

If your child says…“A number is a root if it divides the constant.”

Ask: If substitution leaves unequal sides, can that number be a root?
What’s true: Substitution must make the two sides equal.

$x^2-3x+2=0$

$x^2-x-2x+2=0$

$x(x-1)-2(x-1)=0$

$(x-1)(x-2)=0$

3. Solution of a quadratic equation by factorisation

When a product is zero, set each factor to zero and solve.

Show: Write these lines. Say, “Minus one and minus two add to minus three and multiply to two.”
Ask: Which values make either bracket zero?
Say: The roots are 1 and 2 because either value makes a factor zero.
Used in: Factorising area equations finds rectangular panel dimensions, helping plan cuts.

If your child says…“The roots are minus one and minus two.”

Ask: Does putting minus one into the first bracket make it zero?
What’s true: Solve each bracket equal to zero instead of copying its number.

Rectangle labelled with width x, length x plus one, and area six square centimetres.

4. Problems involving quadratic equations

Turn a situation into an equation, then choose answers that fit.

Show: Look at the diagram. Read the width, length and area labels. Say, “Multiply the sides to find area.”
Ask: What width and length give this area while keeping the length one centimetre greater?
Say: The width is 2 cm and length 3 cm, giving the required area.
Used in: Garden dimensions calculated from area help gardeners plan beds that fit.

If your child says…“Both roots must be possible widths.”

Ask: Could this rectangle have a width of minus three centimetres?
What’s true: An equation's roots still need checking against what the unknown measures.

Predict and choose the roots

$x^2+2x+2=0$

$D=2^2-4(1)(2)$

$D=4-8$

5. Nature of roots

The discriminant distinguishes different, equal or absent real roots.

Show: Name a, b and c as the squared-term coefficient, linear-term coefficient and constant. Call the discriminant D and write $D=b^2-4ac$ . The quadratic formula takes the square root of D. Write these lines.
Ask: What will the next line tell us about real roots?
Say: The discriminant is negative: $D=-4$ , so there are no real roots.
Used in: A discriminant check shows whether given fencing can enclose a requested rectangular area.

If your child says…“I can ignore the minus when taking the square root.”

Ask: Which real number could give minus four when squared?
What’s true: Squaring a real number never produces a negative result.

$x^2+2x+k=0$

$2^2-4(1)k=0$

$4-4k=0$

$4k=4$

6. Condition for equal roots

Choose a coefficient that makes the discriminant zero.

Show: Write these lines. Say, “k is the coefficient we choose. Equal roots need the square-root part to vanish.” Check that the squared-term coefficient stays nonzero.
Ask: What value of k follows from the last line?
Say: Dividing by 4 gives $k=1$ , making the discriminant zero.
Used in: Equal roots identify when a thrown ball reaches its greatest height.

If your child says…“Real roots always need a zero discriminant.”

Ask: When the discriminant is positive, can the formula still give real answers?
What’s true: Real roots occur with either a zero or a positive discriminant.

A real problem

Quadratic equations help locate a GPS receiver.

Bancroft published this positioning method in 1985.

It finds the receiver's position and the difference between its clock and GPS time.

The method reduces the positioning calculation to a quadratic equation.

ESA Navipedia Wikipedia

A handheld GPS receiver resting beside a walking trail.

Concept Lab · Common misconception

The tempting mistake in Quadratic Equations

Open standalone misconception page →

Can you divide away a possible solution?

Wrong

Divide x² = 3x by x and keep only x = 3.

Why it feels right. Cancelling x looks like a quick simplification.

Right

Dividing by x assumes x is nonzero.

Worked example

Original

x² = 3x→x(x − 3) = 0

First factor

x = 0→0 = 0

Second factor

x = 3→9 = 9

Factoring keeps both branches. Dividing by x would lose the branch x = 0.

Write x² − 3x = x(x − 3) = 0.
Check x = 0 and x = 3 in the original equation.

Ask your child six quick questions.

One per step. The answer and the reason are given.

After simplifying $x(x+1)=x^2+2$ , is the equation quadratic?

No. The squared terms cancel, leaving $x-2=0$ .
Is 2 a root of $x^2-x=0$ ?

No. Substitution gives $2=0$ , which is false.
Which root comes from x minus 2?

2. Subtracting 2 from 2 makes that factor zero.
Which root can describe this rectangle's width?

2 cm. The other root is negative, so it cannot measure this side.
How many real roots does this equation have?

None. The formula requires a square root of a negative number.
Besides zero, what discriminant values allow real roots?

Positive values. Their square roots are real, giving two different roots.

15 more questions with answers

Teach Quadratic Equations in six steps.

Why it matters

What to teach

Recognise and solve

Predict and choose the roots

How to teach it

Recognise and solve

1. Quadratic equations in standard form

2. Roots of a quadratic equation

3. Solution of a quadratic equation by factorisation

4. Problems involving quadratic equations

Predict and choose the roots

5. Nature of roots

6. Condition for equal roots

Quadratic equations help locate a GPS receiver.

The tempting mistake in Quadratic Equations

Can you divide away a possible solution?

Ask your child six quick questions.

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