- 1. Sum and Product of Roots
Read the sum and the product of the roots straight off the coefficients of a quadratic, use them backwards to find an unknown coefficient, and build an equation from its roots or from a given sum and product, including surd and fractional cases.
- 2. Symmetric Expressions: Answers Without Solving
Find the sum of the squares of the roots, the sum of their reciprocals, the sum of their ratios and their difference from the coefficients alone, and use a condition on the sum of squares to find an unknown letter.
- 3. New Equations from Old: Related Roots
Build the equation whose roots are alpha plus k, one over alpha, alpha over beta, alpha squared or a multiple of alpha, from the sum and product of the original roots, and use a stated relationship between the two roots to find an unknown letter.
- 4. Real, Equal or Neither? The Discriminant
Use the discriminant b squared minus 4ac to say whether a quadratic has two different real roots, two equal roots or no real roots, and to find the letter that makes the roots equal. When there are no real roots, name the conjugate pair and see why a sum of squares of roots can be negative.
- 5. Ranges of Values: Real Roots and No Real Roots
Find the range of values of a letter for which a quadratic has real roots, no real roots or two different real roots. Real roots means the discriminant is at least zero. Use a sign line when the letter is squared, remove the value that empties the x squared term, and show that the roots are real for every value of the letter.
- 6. Line Meets Parabola: Intersection and Tangency
Find where a line meets a parabola, or two curves meet, by substituting and solving a quadratic, including curves with a fraction. Use the discriminant to count the meeting points, and to find the letter that makes a line a tangent and the point where it touches.
- 7. Putting It All Together: Choosing the Right Tool
Read a question about a quadratic, name which of the six families it belongs to, choose the tool, and check the answer. One equation is asked six ways, an equation with a letter is read off a dial, a value found with the sum and product is checked with the discriminant, and each classic slip is spotted in a written solution.
