Quadratic Formula

Statement

For a quadratic equation \(ax^2+bx+c=0\), the solutions are:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Why it’s true

• The quadratic formula comes from completing the square on the general quadratic equation.
• The expression inside the square root, \(b^2-4ac\), is called the discriminant.
• The discriminant tells us about the type of solutions:
  • \(b^2-4ac>0\): two real solutions.
  • \(b^2-4ac=0\): one real solution (repeated root).
  • \(b^2-4ac<0\): two complex solutions.

Recipe (how to use it)

1. Identify \(a\), \(b\), \(c\) from the quadratic.
2. Calculate discriminant: \(D=b^2-4ac\).
3. Substitute into formula: \((-b \pm \sqrt{D})/(2a)\).
4. Simplify roots.

Spotting it

Use the quadratic formula when a quadratic cannot be factorised easily, or when asked to find exact solutions.

Common pairings

• Discriminant analysis.
• Graphing quadratics (x-intercepts).
• Completing the square for comparison.

Mini examples

1. Given: \(x^2+3x+2=0\).
   Answer: \(a=1,b=3,c=2\). \(x=(-3±√(9-8))/2=(-3±1)/2 → -1,-2.\)
2. Given: \(2x^2-4x-6=0\).
   Answer: \(a=2,b=-4,c=-6\). Discriminant=64. \(x=(4±8)/4→3,-1.\)

Pitfalls

• Forgetting ± gives only one solution instead of two.
• Sign errors when substituting \(b\).
• Square root mistakes with discriminant.

Exam strategy

• Write down \(a,b,c\) first to avoid mistakes.
• Always check discriminant to see how many roots to expect.
• Leave exact answers in surd form if required.

Summary

The quadratic formula gives the exact solutions of any quadratic: \((-b±√(b^2-4ac))/(2a)\).