Quadratics

Question 1

What are the two possible shapes of a parabola

Answer 1

Concave up (happy face) = a > 0 (or positive x2 term) , minimum turning point

Concave down (sad face) = a < 0 (or negative x2 term) , maximum turning point

Question 2

To solve quadratic equations we can use :

Answer 2

• factorisation
• quadratic formula

Question 3

When a quadratic is expressed in the form y=(x+p)^2+q
What is the equation of the axis of symmetry and the tp

Answer 3

• the equation of the axis of symmetry is x= -p
• the turning point is (-p,q)

Question 4

What is the method for completing the square

Answer 4

1. Write an empty bracket being squared with an x inside , look at the coefficient of the x term and half it and write it In the bracket
2. Whatever number you wrote in that bracket , square it and subtract it from the end
3. Simplify the numbers at the end

Question 5

Method for harder completing the square

Answer 5

This is where the coefficient of the x2 term is greater than 1 or when it is negative

Method :
1) take out a common factor
2) complete the square of the quadratic inside the square brackets as normal (keep the common factor outside the bracket)
3) multiply by common factor

Question 6

What is the method of solving a quadratic inequality

Answer 6

1) change the inequality to equals (set to zero)
2) solve the quadratic equation (factorise, quadratic formula, complete the square)
3) plot the roots on a coordinate diagram
4) sketch the graph —> happy face if it’s a positive x2
—> sad face if it’s a negative x2
5) answer the question ==> >0 is where the graph is above the x-axis
<0 is where the graph is below the x-axis

Question 7

If b2-4ac >0

Answer 7

The roots are real and distinct (2 roots)

Question 8

If b2-4ac=0

Answer 8

The roots are real and equal (1root) (ie repeated root)

Question 9

If b2-4ac<0

Answer 9

The roots are not real (0 roots)(ie they don’t exist)

Question 10

How to prove a line is tangent to a curve

Answer 10

• show that b^2- 4ac =0 , one point of intersection , tangent
  OR
• solve quadratic equation showing you only get one solution (repeated root)

Question 11

To show something is always greater than or equal to zero or real

Answer 11

Show you have a squared term