Algebra

When does an equation have real roots?

b² -4ac > 0
crosses the x-axis

When does an equation have no real roots?

b² -4ac < 0
does not cross the x-axis

If x < 0, is it above or below the x axis?

below

If x > 0, is it above or below the x axis?

above

If x > 0, how is the range of values written?

x < y ∪ x > z

If x < 0, how is the range of values written

y < x < z

If asked to prove,
a² -ab +b² ≥ ab
what is the format of the answer?

_² ≥ 0 ✓

If asked to find the range of values for x for which,
x/2x-1 < -2
x ∈ R, what can be done instead of cross multiplication?

multiply by a known positive
(2x-1)² x/2x-1 < -2(2x-1)²
then proceed as normal

Are absolute values + or - ?

unknown

Find range of values for x, where x is less than 1,
| x | < 1

-1 < x < 1
this spread can be used when the coefficient of x is + and when the question contains a less than sign <

Find range of values for x, where x is greater than 1
| x | > 1

square both sides to ensure the number is positive,
x < -1 ∪ x > 1

When changing the signs in an equation, what happens to the greater than or less than symbol?

it also changes

What shape are absolute value graphs?

they are initally an X shape, but become a V shape

How would you graph,
y = | x -3 |

we do not know if | x -3 | is positive of negative, and must account for both
first assume it is positive, second assume it is negative
use y = mx +c to calculate the points on the graph

What is the formula for forming a quadratic equation when given the roots?

y = x² - (sum of the roots)x + product of the roots

How do you find the coefficient of x^number in a binomial expansion?

let the expansion/term being expanded = T_r+1

T_r+1 = Coefficentx^number

Expand term using r

Solve for r using the power of x

or

Cross multiply both terms containing x^number