Quadratic Equations: SPI Mathematics

Formulas (2)

• Standard form of a quadratic equation · Standard form
  $$ax^2 + bx + c = 0, \quad a \ne 0$$
• Writing a word situation as a quadratic equation · Pythagoras theorem
  $$(\text{hypotenuse})^2 = (\text{base})^2 + (\text{height})^2$$

Watch out for (6)

• Thinking b or c must be non-zero
  → Standard form of a quadratic equation
• Keeping the sign when a term crosses the = sign
  → Standard form of a quadratic equation
• Judging by the look before expanding
  → Simplifying both sides before naming the type of equation
• Confusing a root in the coefficient with a root of x
  → Simplifying both sides before naming the type of equation
• Adding the hypotenuse to another side
  → Writing a word situation as a quadratic equation
• Reading 'more than' backwards
  → Writing a word situation as a quadratic equation

Formulas (3)

• A root of a quadratic equation makes it true · Condition for a root
  $$a\alpha^2 + b\alpha + c = 0$$
• Solving a quadratic equation by splitting the middle term · Zero product rule
  $$(x - p)(x - q) = 0 \Rightarrow x = p \text{ or } x = q$$
• The quadratic formula · Quadratic formula
  $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Watch out for (6)

• Squaring a negative number wrongly
  → A root of a quadratic equation makes it true
• Dividing both sides by x
  → Solving a quadratic equation by splitting the middle term
• Taking the root with the bracket's sign
  → Solving a quadratic equation by splitting the middle term
• Factorising before one side is 0
  → Solving a quadratic equation by splitting the middle term
• Losing the sign of b
  → The quadratic formula
• Dividing only part of the top by 2a
  → The quadratic formula

Formulas (3)

• Nature of the roots from the discriminant · Discriminant
  $$D = b^2 - 4ac$$
• Values of k that give real, distinct or no real roots · Conditions on the discriminant
  $$b^2 - 4ac \ge 0 \text{ (real roots)} \qquad b^2 - 4ac > 0 \text{ (distinct roots)} \qquad b^2 - 4ac < 0 \text{ (no real roots)}$$
• Equal roots: setting the discriminant to zero · Equal roots
  $$b^2 - 4ac = 0 \qquad x = -\frac{b}{2a}$$

Watch out for (7)

• Dropping the sign of c
  → Nature of the roots from the discriminant
• Calling equal roots rational
  → Nature of the roots from the discriminant
• Keeping only one side of a square inequality
  → Values of k that give real, distinct or no real roots
• Using > when the question says real
  → Values of k that give real, distinct or no real roots
• Taking only the positive square root
  → Equal roots: setting the discriminant to zero
• Reading b before expanding
  → Equal roots: setting the discriminant to zero
• Ignoring 'positive' in 'equal and positive roots'
  → Equal roots: setting the discriminant to zero

Formulas (3)

• Sum and product of the roots from the coefficients · Sum and product of roots
  $$\alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a}$$
• Forming a quadratic equation from its roots · Equation from its roots
  $$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$
• Rebuilding an equation from wrongly copied coefficients · Correct equation from the two students' trusted values
  $$x^2 - (\text{sum from the student with the right } x \text{ term})\,x + (\text{product from the student with the right constant}) = 0$$

Watch out for (5)

• Forgetting the minus in the sum
  → Sum and product of the roots from the coefficients
• Not dividing by a
  → Sum and product of the roots from the coefficients
• Writing plus the sum
  → Forming a quadratic equation from its roots
• Leaving fractions in the equation
  → Forming a quadratic equation from its roots
• Trusting the wrong student's sum
  → Rebuilding an equation from wrongly copied coefficients