2. Quadratics Flashcards
Method for ax^2 + bx + c when a isn’t 1
- Multiply c by a
- Find the two numbers that add to b and multiply to make c (d and e)
- Write as ax^2 + dx + ex + original c
- Factorise from that
Difference of two cubes
For equation a^3 - b^3:
a-b)(a^2 + ab + b^2
Final step when factorising
Check that none of the brackets can be further factorised or simplified
x(a+b) + y(a+b)^2
(a + b)(x + y(a + b))
a + b)(x + ya + yb
x(x-y) + y(x-y)
(x-y)(x+y)
Express 2x^2 + 12x + 7 in a(x+b)^2 + c
2(x^2 + 6x) - take out coefficient of x^2 from x and x^2
2(x+3)^2 - make first bracket for that
2[(x^2 + 6x + 9)] - factorise
2[(x^2 + 6x + 9) - 9] + 7 take away the number
2(x+3)^2 -18 + 7 - expand
2(x+3)^2 - 11 - solve
Features needed in a sketch
- Roots (factorise and solve)
- y-intercept (substitute x = 0)
- Turning point (complete the square (-b,c))
- Line of symmetry (x value of completed square)
Proving the quadratic formula pt1
ax^2 + bx + c = 0
x^2 + b/a x + c/a = 0 - divide all by a
x^2 + b/a x = - c/a
(x + b/2a)^2 = -c/a + (b/2a)^2
Proving the quadratic formula pt. 2
(x + b/2a)^2 = (b2-4ac)/4a^2 -simplify RHS
x + b/2a = (square root of b2-4ac/2a) - square root everything
x = -b/2a +- square root of b2-4ac/2a
Simplify fractions to give you formula
Discriminant
b^2 - 4ac
if b^2 - 4ac > 0
Two real roots
if b^2 - 4ac = 0
Equal roots
if b^2 - 4ac < 0
No real roots
Questions where you know an equation with unknowns has equal roots
b^2 - 4ac = 0
How to determine an equation from the graph
- Take the roots
- Make the brackets (x-root1)(x-root2)
- Expand
- If c from the expansion doesn’t equal the y intercept put a factor in front so it does
