THE COMPOUND // FIELD NOTES // WEEK 13
Radical equations, completing the square, and complex numbers
Field Book
Lessons 48 to 51
Due
Test 11 due ยท Supplement 13
Method
Paper first, every step
Two doors open this week: completing the square, and the number i. Both stay open for the rest of the year.
What you have to be able to do
- Solve radical equations by isolating and squaring
- Solve quadratics by completing the square
- Compute with imaginary and complex numbers
The procedures
Radical equationsField Book Lesson 48
- Isolate the radical on one side, then square both sides.
- Check every answer in the original: squaring can invent solutions.
Equation of a line from its graphField Book Lesson 49
- Read the intercept b straight off the crossing.
- Count rise over run for m, then write y = mx + b.
Completing the squareField Book Lesson 50
- Move the constant across. Take half the x coefficient, square it, add to both sides.
- Write the perfect square, then take both square roots and finish.
Imaginary and complex numbersField Book Lesson 51
- i is the square root of -1, so i2 = -1.
- A complex number is a + bi. Combine like parts: reals with reals, imaginaries with imaginaries.
Worked: completing the square
- Solve x2 + 6x + 5 = 0.
- x2 + 6x = -5. Half of 6 is 3, squared is 9: x2 + 6x + 9 = 4.
- (x + 3)2 = 4, so x + 3 = 2 or x + 3 = -2. x = -1 or x = -5.
Terms on the record
| Term | Meaning |
|---|
| radical equation | an equation with the variable under a radical |
| completing the square | adding the square of half the linear coefficient to build a perfect square |
| imaginary number | a multiple of i, where i squared is negative one |