Quadratics and Polynomials

Question 1

Describe the three rules for the nature of the discriminant.

Answer 1

If discriminant > 0, real and distinct roots, if < 0, no real roots, if = 0, real and equal roots.

Question 2

Completing The Square (Hard Version)

Answer 2

Take out the coefficient of x as a factor, square bracket off the x terms, work inside the bracket, and at the end, you times inside the square bracket by the factor, and tidy up accordingly.

Question 3

Sketching Parabolas

Answer 3

Check the discriminant to see if it can be factorised, if there are roots, factorise for x values. If not, complete the square and sketch. To get x, make y=0, to get y, make x=0, get x axis symmetry, sub in for turning point value and annotate.

Question 4

Determine the Equation of a Curve when given points.

Answer 4

Write out roots and get factors. Put in form y=k(x-a)(x-b)(x-c). Sub in y intercept points to get the value of k. Rewrite and multiply out in full.

Question 5

Write out roots and get factors. Put in form y=k(x-a)(x-b)(x-c). Sub in y intercept points to get the value of k. Rewrite and multiply out in full.

Answer 5

Get x values, sketch, if it’s looking for > 0, look to above x axis and vice versa. If it’s the curve, you put x in the middle and numbers on either side. If it’s two ends, you take the outside values for it. Write mad way. Remember concave upwards or downwards.

Question 6

Showing a Line is Tangent to a Parabola

Answer 6

Solve this by making them equal to eachother for x value, one repeated root means it is a tangent. Sub in the x for the y value and write as a co-ordinate for the point of contact.

Question 7

Synthetic Division

Answer 7

Remainder is what’s left after solving, quotient is the numbers taken underneath and dropped to a power. Solving fully uses these to get the three x values (or more). To get values of p etc you may need to work algebraically.