Quadratic Equations

The standard shape

• A quadratic equation contains a squared term and nothing of higher power, and its standard shape is ax² + bx + c = 0, where a is not zero
• All three methods below need zero on one side, so rearranging comes before choosing a method
• Collect everything on the side that leaves the x² term positive; if the squared term ends up negative, multiply every term by −1
  • 12 − x² = 4x becomes 0 = x² + 4x − 12, which is easier to work with than −x² − 4x + 12 = 0
• The rearranged equation has three terms, and reaching it correctly is worth a mark on its own even before any solving happens
• Most quadratics give two solutions; some give one repeated value, and a few have none

Which method to use

• Factorising is quickest whenever the expression factorises with whole numbers
• The formula always works, and is the method to reach for when the numbers are awkward or the answer is wanted as a decimal
• Completing the square always works too, and is the method asked for when the question wants a squared bracket or a turning point
• A question that says "solve" without naming a method leaves the choice to you, so a failed attempt at factorising costs only time, not marks