Quadratics

Explanation

A connected toolkit for section 1.1

A quadratic polynomial has the form ax^2 + bx + c with a non-zero. The same expression can be written in expanded form, factorised form or completed-square form. Each form reveals different information: expanded form exposes the coefficients, factorised form exposes roots, and completed-square form exposes the vertex and range.

Before calculating, decide what the question asks you to reveal. Factorisation is efficient when integer or simple rational factors exist; completing the square is especially useful for a vertex, a range or an exact equation; the quadratic formula works for every quadratic equation. A final answer should respect any stated domain, interval, physical restriction or rounding instruction.

This pack is limited to Pure Mathematics 1 section 1.1. It uses elementary algebra and, in one disguised-equation discussion, familiar function notation, but it does not claim to cover the rest of Paper 1 or a complete Cambridge International AS Level route.

Explanation

Completing the square

For x^2 + bx, half the coefficient of x and square it: x^2 + bx = (x + b/2)^2 - (b/2)^2. Hence x^2 + bx + c = (x + b/2)^2 + c - b^2/4. Expanding the completed square is a reliable check.

When a is not 1, first factor a from both terms containing x: ax^2 + bx + c = a[x^2 + (b/a)x] + c. Complete the square inside the bracket and remember that the correction term is multiplied by a. Equivalently, ax^2 + bx + c = a(x + b/(2a))^2 + c - b^2/(4a).

Because a square is non-negative for real x, the completed-square form also gives an extremum. If a is positive, the vertex is a minimum; if a is negative, it is a maximum.

Explanation

Using completed-square form for graph information

In y = a(x - h)^2 + k, the vertex is (h, k) and the axis of symmetry is x = h. Notice the sign: (x + 4)^2 means h = -4. The sign of a tells whether the graph opens upward or downward, while |a| affects its steepness.

A sketch can be planned without plotting many points. Mark the vertex, the axis, the y-intercept found by setting x = 0, and any real x-intercepts found by setting y = 0. If there are no real x-intercepts, the entire graph remains on one side of the x-axis.

When the domain is restricted, the range may no longer start or end at the unrestricted vertex. Check whether the vertex's x-coordinate belongs to the stated domain; if it does not, compare the values permitted at the domain boundary or endpoints.

Explanation

The discriminant and the number of real roots

For ax^2 + bx + c = 0, the discriminant is D = b^2 - 4ac. It is the expression beneath the square root in the quadratic formula. If D > 0 there are two distinct real roots; if D = 0 there is one repeated real root; if D < 0 there are no real roots.

The discriminant also describes contact between y = ax^2 + bx + c and the x-axis. Two roots mean two crossings, a repeated root means the graph touches the axis at its vertex, and no real roots mean no contact. This conclusion concerns real roots only.

In parameter problems, substitute the coefficients carefully and solve an inequality or equation in the parameter. Pay attention when dividing an inequality by a negative number, because its direction reverses.

Explanation

Choosing a method for a quadratic equation

First rearrange to ax^2 + bx + c = 0. Factorise when a product of linear factors is visible, then use the zero-product rule. Completing the square is useful when the equation is close to a perfect square or when its structure matters. The formula x = [-b +/- sqrt(b^2 - 4ac)]/(2a) is universal when a is non-zero.

Keep exact values such as (5 + sqrt(13))/2 unless a decimal is requested. If rounding is requested, retain extra calculator digits until the final line and state both roots to the required accuracy. A decimal instruction such as '3 significant figures' is part of the answer, not an optional presentation choice.

Always check whether algebra has introduced candidates that the original problem forbids. Pure polynomial manipulation does not normally create extraneous roots, but squaring, clearing a variable denominator or reversing a substitution can do so.

Explanation

Quadratic inequalities

To solve a quadratic inequality, move every term to one side and find the critical values where the quadratic equals zero. These roots divide the number line into intervals on which the sign cannot change. Determine the sign in each interval from the graph's orientation or by testing one value.

For a positive leading coefficient and two real roots r1 < r2, the quadratic is positive outside the roots and negative between them. For a negative leading coefficient the signs reverse. Include a root for <= or >=, but exclude it for < or >.

Do not manipulate a quadratic inequality as if it were a linear equation by taking square roots without considering signs. A sign chart or parabola argument keeps disconnected solution intervals and endpoints visible.

Explanation

Linear-quadratic simultaneous equations

When one equation is linear and the other quadratic, rearrange the linear equation to express one variable in terms of the other. Substitute into the quadratic equation, solve the resulting quadratic, and then recover the corresponding second coordinate for each root.

A quadratic may give zero, one or two real intersection points. Keep the pairing correct: each x-value must be matched with the y-value obtained from the same substitution. Check every ordered pair in both original equations.

If the algebra produces no real roots, the two real graphs do not intersect. A repeated root represents one point of contact. These interpretations require no diagram, and every question in this pack supplies all needed equations in text.

Explanation

Recognising a disguised quadratic

An equation is disguised quadratic when the same expression appears squared and to the first power. For example, x^4 - 5x^2 + 4 = 0 is quadratic in x^2. Set u = x^2, solve u^2 - 5u + 4 = 0, and then reverse the substitution.

The reverse step has its own domain. If u = x^2 for real x, then u must be non-negative and each positive u gives two x-values. If u = sqrt(x), then u >= 0 and x = u^2. If u is a reciprocal, excluded denominator values from the original equation remain excluded.

After reversing a substitution, check every candidate in the original equation. This is essential when squaring was used, because squaring can turn two unequal expressions with opposite signs into equal squares and create an extraneous root.

Revision checklist and common errors

Completed square: factor out a when needed, halve the inner x-coefficient, and verify by expansion. Read the vertex from a(x - h)^2 + k as (h, k), then use the stated domain before declaring a range.

Discriminant and equations: compute b^2 - 4ac with brackets around negative coefficients. Distinguish D > 0, D = 0 and D < 0. Choose an efficient method, preserve exact forms unless decimals are requested, and round only on the final line.

Inequalities and simultaneous equations: put the quadratic on one side, find boundary roots, determine interval signs and treat endpoints correctly. In a linear-quadratic pair, substitute, recover every matching coordinate and check each ordered pair in both equations.

Disguised quadratics: define the temporary variable clearly, record its domain, solve the quadratic, reverse the substitution and reject invalid or extraneous candidates. Excluded scope: this pack does not cover the other Pure Mathematics 1 sections, other components, or complete AS Mathematics.