Quadratic Equations — CDS Elementary Mathematics

Formulas (2)

• Forming an equation from its roots · Equation from sum and product
  $$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$
• When the coefficients add to zero · Root 1
  $$a + b + c = 0 \;\Rightarrow\; x = 1,\ \dfrac ca$$

Watch out for (2)

• The sign of the sum
  → Forming an equation from its roots
• Clear the fractions first
  → When the coefficients add to zero

Formulas (3)

• Squares, reciprocals and products of the roots · Sum of squares of the roots
  $$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$$
• The difference of the roots · Difference of the roots
  $$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$$
• Cubes of the roots · Sum of cubes of the roots
  $$\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$$

Watch out for (3)

• A bound that is never reached
  → Squares, reciprocals and products of the roots
• The difference fixes k only up to sign
  → The difference of the roots
• Keep a leading coefficient
  → Cubes of the roots

Formulas (2)

• Roots in a ratio · Roots in the ratio m : n
  $$mn\,b^2 = (m + n)^2\,ac$$
• When the roots are the coefficients · Roots p and q of x² + px + q = 0
  $$p + q = -p, \quad pq = q \;\Rightarrow\; (p, q) = (0, 0) \text{ or } (1, -2)$$

Watch out for (2)

• Keep both signs of t
  → Roots in a ratio
• Do not divide by a letter that can be zero
  → When the roots are the coefficients

Formulas (2)

• Real, equal or no real roots · Discriminant
  $$D = b^2 - 4ac$$
• Ranges and counts from the discriminant · Real roots
  $$b^2 - 4ac \ge 0$$

Watch out for (2)

• k = 0 can be a valid answer
  → Real, equal or no real roots
• For every θ means the worst θ
  → Ranges and counts from the discriminant

Formulas (2)

• Perfect squares and expressions of one sign · Perfect square
  $$ax^2 + bx + c = a\left(x + \tfrac{b}{2a}\right)^2 \iff b^2 = 4ac$$
• Signs, integer roots and roots between two numbers · A point between the roots
  $$a > 0,\ f(k) < 0 \;\Rightarrow\; \alpha < k < \beta$$

Watch out for (2)

• A perfect square can be zero somewhere
  → Perfect squares and expressions of one sign
• Opposite signs does not need D > 0 separately
  → Signs, integer roots and roots between two numbers

Formulas (2)

• Subtract to find the common root · Common root by subtraction
  $$(x^2 + px + q) - (x^2 + qx + p) = (p - q)(x - 1)$$
• Factor one and try its roots · Common root from a factorised equation
  $$(x - r_1)(x - r_2) = 0 \;\Rightarrow\; \text{common root is } r_1 \text{ or } r_2$$

Watch out for (2)

• Discard the value the stem rules out
  → Subtract to find the common root
• Two answers, not one
  → Factor one and try its roots

Formulas (2)

• Completing the square · Vertex of a quadratic
  $$x = -\dfrac{b}{2a}, \qquad \text{extreme value} = \dfrac{4ac - b^2}{4a}$$
• Extremes of a reciprocal · Greatest value of a reciprocal
  $$\max \dfrac{1}{ax^2 + bx + c} = \dfrac{4a}{4ac - b^2} \quad (a > 0,\ b^2 < 4ac)$$

Watch out for (2)

• Take out a before completing
  → Completing the square
• Invert the value, not the expression
  → Extremes of a reciprocal

Formulas (2)

• Substitute to get a quadratic · Reciprocal substitution
  $$t = x - \tfrac1x \;\Rightarrow\; x^2 + \tfrac{1}{x^2} = t^2 + 2$$
• Clear roots and fractions, then check · Squaring a root equation
  $$\sqrt{f(x)} = g(x) \iff f(x) = g(x)^2 \text{ and } g(x) \ge 0$$

Watch out for (2)

• Some roots of the new quadratic give no real x
  → Substitute to get a quadratic
• Squaring adds roots
  → Clear roots and fractions, then check

Formulas (2)

• Number problems · Sum of squares of consecutive numbers
  $$n^2 + (n + 1)^2 = 2n^2 + 2n + 1$$
• Lengths and prices · Same total, two prices
  $$\dfrac{T}{n} - \dfrac{T}{n + k} = r \;\Rightarrow\; n(n + k) = \dfrac{kT}{r}$$

Watch out for (2)

• Both roots can be right
  → Number problems
• Keep the root inside the segment
  → Lengths and prices