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Quadratic Equations formulas

19 formulas and 19 common traps for CDS Mathematics Quadratic Equations, grouped by subtopic.

Full notes with worked examples

Solving and Forming Quadratic Equations

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Forming an equation from its roots

Equation from sum and product

x2−(α+β)x+αβ=0x^2 - (\alpha + \beta)x + \alpha\beta = 0

When the coefficients add to zero

Root 1

a+b+c=0  ⇒  x=1, caa + b + c = 0 \;\Rightarrow\; x = 1,\ \dfrac ca

Common traps

The sign of the sum

The xx-coefficient is MINUS the sum. Roots with sum 22 give x2−2x+…x^2 - 2x + \ldots, not x2+2x+…x^2 + 2x + \ldots, and the wrong-sign version is always among the options.

Clear the fractions first

The coefficient-sum test works on the equation as it stands, fractions and all, but the product ca\dfrac ca is easier to read after multiplying through. Do not multiply only some of the terms.

Symmetric Functions of the Roots

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Squares, reciprocals and products of the roots

Sum of squares of the roots

α2+β2=(α+β)2−2αβ\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta

The difference of the roots

Difference of the roots

(α−β)2=(α+β)2−4αβ(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta

Cubes of the roots

Sum of cubes of the roots

α3+β3=(α+β)3−3αβ(α+β)\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)

Common traps

A bound that is never reached

When a parameter varies, an expression like 18k−2\dfrac{18}{k} - 2 with k<0k < 0 gets as close to −2-2 as you like but never equals it. Check whether the extreme value is actually attained before calling it the maximum.

The difference fixes k only up to sign

k2=49k^2 = 49 gives k=±7k = \pm 7; the question usually says which sign it wants. Squared conditions always lose the sign.

Keep a leading coefficient

For 2x2−…2x^2 - \ldots the product is the constant DIVIDED BY 22. Forgetting to divide is the commonest slip on this page, and it produces an answer twice too large in one term.

Roots in a Given Relation

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Roots in a ratio

Roots in the ratio m : n

mn b2=(m+n)2 acmn\,b^2 = (m + n)^2\,ac

When the roots are the coefficients

Roots p and q of x² + px + q = 0

p+q=−p,pq=q  ⇒  (p,q)=(0,0) or (1,−2)p + q = -p, \quad pq = q \;\Rightarrow\; (p, q) = (0, 0) \text{ or } (1, -2)

Common traps

Keep both signs of t

t2t^2 from the product gives two values of tt, and each gives a different coefficient. The question usually asks for 'the positive value'; make sure you pick the tt that produces it.

Do not divide by a letter that can be zero

From pq=qpq = q you may conclude p=1p = 1 only when q≠0q \ne 0. If the question does not say so, p=q=0p = q = 0 is also a solution, and an option like 'p=0p = 0 or 11' becomes the correct one.

Nature of Roots and the Discriminant

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Real, equal or no real roots

Discriminant

D=b2−4acD = b^2 - 4ac

Ranges and counts from the discriminant

Real roots

b2−4ac≥0b^2 - 4ac \ge 0

Common traps

k = 0 can be a valid answer

A value of kk that makes the equation 4x2=04x^2 = 0 still gives equal roots (both zero). Do not throw it away unless it kills the x2x^2 term.

For every θ means the worst θ

If p≤14cos⁡2θp \le \dfrac14\cos^2\theta must hold for every θ\theta in a range, pp is limited by the SMALLEST value of cos⁡2θ\cos^2\theta there, not the largest.

Perfect Squares, Signs and Location of Roots

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Perfect squares and expressions of one sign

Perfect square

ax2+bx+c=a(x+b2a)2  ⟺  b2=4acax^2 + bx + c = a\left(x + \tfrac{b}{2a}\right)^2 \iff b^2 = 4ac

Signs, integer roots and roots between two numbers

A point between the roots

a>0, f(k)<0  ⇒  α<k<βa > 0,\ f(k) < 0 \;\Rightarrow\; \alpha < k < \beta

Common traps

A perfect square can be zero somewhere

(x−2)2(x - 2)^2 is a perfect square but not positive for EVERY xx: it is 00 at x=2x = 2. 'Positive for all xx' needs D<0D < 0, strictly.

Opposite signs does not need D > 0 separately

If ca<0\dfrac ca < 0, then b2−4ac>0b^2 - 4ac > 0 automatically, since −4ac-4ac is positive. A statement giving D>0D > 0 adds nothing to a product condition.

Common Roots

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Subtract to find the common root

Common root by subtraction

(x2+px+q)−(x2+qx+p)=(p−q)(x−1)(x^2 + px + q) - (x^2 + qx + p) = (p - q)(x - 1)

Factor one and try its roots

Common root from a factorised equation

(x−r1)(x−r2)=0  ⇒  common root is r1 or r2(x - r_1)(x - r_2) = 0 \;\Rightarrow\; \text{common root is } r_1 \text{ or } r_2

Common traps

Discard the value the stem rules out

The subtraction often gives two candidates, one of which makes a parameter zero or makes the two equations identical. If the stem says k≠0k \ne 0 or p≠qp \ne q, that candidate is gone.

Two answers, not one

Each root of the factorised equation can be the shared one, so there are usually two values of the unknown. An option giving only one of them is incomplete.

Maximum and Minimum of a Quadratic

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Completing the square

Vertex of a quadratic

x=−b2a,extreme value=4ac−b24ax = -\dfrac{b}{2a}, \qquad \text{extreme value} = \dfrac{4ac - b^2}{4a}

Extremes of a reciprocal

Greatest value of a reciprocal

max⁡1ax2+bx+c=4a4ac−b2(a>0, b2<4ac)\max \dfrac{1}{ax^2 + bx + c} = \dfrac{4a}{4ac - b^2} \quad (a > 0,\ b^2 < 4ac)

Common traps

Take out a before completing

In 2x2+8x+12x^2 + 8x + 1, factor 22 from the xx terms first: 2(x+2)2−72(x + 2)^2 - 7. Completing x2+8xx^2 + 8x inside without the 22 gives the wrong vertex.

Invert the value, not the expression

The least denominator is 74\dfrac74 for x2+3x+4x^2 + 3x + 4, so the greatest fraction is 47\dfrac47. An option 74\dfrac74 is the denominator's value, not the answer.

Equations Reducible to Quadratics

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Substitute to get a quadratic

Reciprocal substitution

t=x−1x  ⇒  x2+1x2=t2+2t = x - \tfrac1x \;\Rightarrow\; x^2 + \tfrac{1}{x^2} = t^2 + 2

Clear roots and fractions, then check

Squaring a root equation

f(x)=g(x)  ⟺  f(x)=g(x)2 and g(x)≥0\sqrt{f(x)} = g(x) \iff f(x) = g(x)^2 \text{ and } g(x) \ge 0

Common traps

Some roots of the new quadratic give no real x

u=x2u = x^2 must be non-negative, and t=x+1xt = x + \dfrac1x must satisfy ∣t∣≥2|t| \ge 2. A root of the uu or tt equation outside that range is dropped.

Squaring adds roots

Both f=g\sqrt{f} = g and f=−g\sqrt{f} = -g square to the same equation. Every root of the squared equation must be tried in the original; count only those that pass.

Word Problems with Quadratics

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Number problems

Sum of squares of consecutive numbers

n2+(n+1)2=2n2+2n+1n^2 + (n + 1)^2 = 2n^2 + 2n + 1

Lengths and prices

Same total, two prices

Tn−Tn+k=r  ⇒  n(n+k)=kTr\dfrac{T}{n} - \dfrac{T}{n + k} = r \;\Rightarrow\; n(n + k) = \dfrac{kT}{r}

Common traps

Both roots can be right

'The sum of a number and its square is 3030' has two answers, 55 and −6-6, because nothing says the number is positive. Reject a root only when the stem rules it out.

Keep the root inside the segment

When a point CC divides a segment of length 44, a root like 6+256 + 2\sqrt5 is longer than the segment itself. Only the root between 00 and the length is a real position.

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