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Core notes

Equations identify values that satisfy stated relationships, while inequalities identify intervals. The discriminant classifies quadratic roots and line-curve intersections, substitution reduces a simultaneous system to one variable, and sign analysis turns quadratic factors into a number-line solution.

Discriminants and line-curve intersections

For $ax^2+bx+c=0$ , the discriminant $\Delta=b^2-4ac$ gives two distinct real roots when $\Delta>0$ , one repeated real root when $\Delta=0$ , and no real roots when $\Delta<0$ . To classify how a line meets a curve, solve the two equations simultaneously until one quadratic remains, then apply the same test.

Simultaneous equations by substitution

Rearrange the linear equation to express one variable in terms of the other, substitute into the second equation, and solve the resulting equation. A quadratic result may produce two ordered pairs. Substitute each pair back into both original equations.

Quadratic inequalities

Move every term to one side and find the real roots of the corresponding quadratic equation. These roots divide the number line into intervals. Determine the sign in each interval from the graph, factors, or a test value, then include a boundary only when the inequality permits equality.

Relationship

Use

\Delta=b^2-4ac

Classify the real roots of a quadratic.

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Solve a quadratic exactly when factorisation is unsuitable.

Worked examples

Example 1: Find the values of $k$ for which the line $y=kx+1$ is tangent to $y=x^2+3$ .

Equate the expressions: $x^2+3=kx+1$ , so $x^2-kx+2=0$ .
Tangency requires one repeated root, so $\Delta=0$ .
Solve $k^2-8=0$ .

Answer: $k=\pm2\sqrt{2}$ .

Example 2: Solve $x^2-5x+6\leq0$ .

Factorise: $(x-2)(x-3)\leq0$ .
The roots are $2$ and $3$ ; the upward-opening graph is at or below the axis between them.
Include both roots because equality is allowed.

Answer: $2\leq x\leq3$ .

Exam traps and retrieval check

Avoid these traps

Applying a discriminant before reducing the relation to a single quadratic.
Keeping only one solution from a simultaneous system that produces two valid ordered pairs.
Reversing the inside and outside intervals of a quadratic inequality.

Check from memory

What discriminant condition represents tangency?

$\Delta=0$ .

Why must simultaneous-equation answers be checked in both originals?

Substitution and later algebra can introduce or retain values that do not satisfy the full system.

When is an inequality boundary included?

When the symbol is $\leq$ or $\geq$ and the boundary is in the stated domain.

Core notes

Discriminants and line-curve intersections

Simultaneous equations by substitution

Quadratic inequalities

Relationship

Use

\Delta=b^2-4ac

Classify the real roots of a quadratic.

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Solve a quadratic exactly when factorisation is unsuitable.

Worked examples

Example 1: Find the values of $k$ for which the line $y=kx+1$ is tangent to $y=x^2+3$ .

Equate the expressions: $x^2+3=kx+1$ , so $x^2-kx+2=0$ .
Tangency requires one repeated root, so $\Delta=0$ .
Solve $k^2-8=0$ .

Answer: $k=\pm2\sqrt{2}$ .

Example 2: Solve $x^2-5x+6\leq0$ .

Factorise: $(x-2)(x-3)\leq0$ .
The roots are $2$ and $3$ ; the upward-opening graph is at or below the axis between them.
Include both roots because equality is allowed.

Answer: $2\leq x\leq3$ .

Exam traps and retrieval check

Avoid these traps

Applying a discriminant before reducing the relation to a single quadratic.
Keeping only one solution from a simultaneous system that produces two valid ordered pairs.
Reversing the inside and outside intervals of a quadratic inequality.

Check from memory

What discriminant condition represents tangency?

$\Delta=0$ .

Why must simultaneous-equation answers be checked in both originals?

Substitution and later algebra can introduce or retain values that do not satisfy the full system.

When is an inequality boundary included?

When the symbol is $\leq$ or $\geq$ and the boundary is in the stated domain.

A2: Equations and inequalities

Core notes

Discriminants and line-curve intersections

Simultaneous equations by substitution

Quadratic inequalities

Formulae and methods

Worked examples

Example 1: Find the values of $k$ for which the line $y=kx+1$ is tangent to $y=x^2+3$ .

Example 2: Solve $x^2-5x+6\leq0$ .

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

A2: Equations and inequalities

Core notes

Discriminants and line-curve intersections

Simultaneous equations by substitution

Quadratic inequalities

Formulae and methods

Worked examples

Example 1: Find the values of $k$ for which the line $y=kx+1$ is tangent to $y=x^2+3$ .

Example 2: Solve $x^2-5x+6\leq0$ .

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Relationship	Use
$\Delta=b^2-4ac$	Classify the real roots of a quadratic.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$	Solve a quadratic exactly when factorisation is unsuitable.

Core notes

Discriminants and line-curve intersections

Simultaneous equations by substitution

Quadratic inequalities

Formulae and methods

Worked examples

Example 1: Find the values of kkk for which the line y=kx+1y=kx+1y=kx+1 is tangent to y=x2+3y=x^2+3y=x2+3.

Example 2: Solve x2−5x+6≤0x^2-5x+6\leq0x2−5x+6≤0.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Core notes

Discriminants and line-curve intersections

Simultaneous equations by substitution

Quadratic inequalities

Formulae and methods

Worked examples

Example 1: Find the values of kkk for which the line y=kx+1y=kx+1y=kx+1 is tangent to y=x2+3y=x^2+3y=x2+3.

Example 2: Solve x2−5x+6≤0x^2-5x+6\leq0x2−5x+6≤0.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Example 1: Find the values of $k$ for which the line $y=kx+1$ is tangent to $y=x^2+3$ .

Example 2: Solve $x^2-5x+6\leq0$ .

Example 1: Find the values of $k$ for which the line $y=kx+1$ is tangent to $y=x^2+3$ .

Example 2: Solve $x^2-5x+6\leq0$ .