Quadratics and Cubics

Question 1

What is the general form of a quadratic equation?

Answer 1

ax^2 + bx + c where a, b and c are constants and a is not 0.

Question 2

How do you factorise a quadratic when a = 1?

Answer 2

• Rearrange to ensure it’s in standard form
• Find the two numbers that multiply together to make c but which also add or subtract to give b.
• Find the signs and put them in the brackets.

Question 3

How do you find the solutions of quadratics that don’t factorise?

Answer 3

Using the quadratic formula:

x = (-b +/- (2)route(b^2 - 4ac))/2a

Question 4

What is completing the square?

Answer 4

Writing a quadratic in the form a(x + b)^2 + d.

Question 5

How do you complete the square when a = 1?

Answer 5

• Write out (x + b/2)^2
• Multiply out the square to find the new c value
• Take away the new c value and add the correct one.

Question 6

How do you factorise a quadratic when a in not 1?

Answer 6

• Times a by c.
• Find the factors of ac which give add or minus to give b.
• Write out the equation as ax^2 + bx + dx +c where b and d are the factors.
• Factorise the two halves of the equation so that the brackets are the same.
• Take the brackets out of both terms.

Question 7

How do you complete the square when a is not 1?

Answer 7

-Take a out so that b becomes b/a and c becomes c/a.
-Complete the square inside the brackets.
times the value outside the square by a in order to remove it from the outer brackets.

Question 8

How do you solve hidden quadratics?

Answer 8

Substitute y as something that will put the equation into the standard form and solve.

Question 9

What are the roots of a quadratic function?

Answer 9

The points at which the quadratic graph crosses the x-axis.

Question 10

What is the shape of a quadratic graph?

Answer 10

If a is positive it will be a u shape and if it’s negative it will be an n shape.

Question 11

What does the completed square function tell you about a quadratic?

Answer 11

For p(x+q)^2 + r:

• If p is positive, the graph will be u-shaped and have a minimum
• if p is negative, the grave will be n-shaped and have a maximum.
• the minimum or maximum is (-q,r) so there is a line of symmetry at x = -q.
• If p and r have different signs, the function has two real roots.
• If r = 0 then the function has one real root.
• if p and r have the same sign, the function has no real roots.

Question 12

What is the discriminant?

Answer 12

The part of the quadratic formula under the route:

b^2 - 4ac

Question 13

What can the discriminant tell us about the quadratic?

Answer 13

If it is positive then you get two real solutions, if it is negative then you get no real solutions, if it is equal to 0 then you get 1 real solution.

Question 14

How do you sketch a quadratic graph?

Answer 14

• If the coefficient of x^2 is positive the graph will be u-shaped. If it is negative it will be n-shaped.
• To find the y-intercept, let x = 0 and calculate the value of y.
• To find the x-intercept, let y = 0 and solve the equation 0 = ax^2 + bx + c to find the value or values of x.

Question 15

How do you find the minimum/maximum of the graph?

Answer 15

• Complete the square to put the graph in the form y = p(x + q) + r.
• The point will be (-q,r).
• If p is positive, it is a minimum. If it is negative then it is a maximum.

Question 16

How can you work out the minimum/maximum if you already know the real roots?

Answer 16

• If the function has two distinct roots, then the vertex will have an x coordinate that is the same as the midpoint between the two roots.
• If the function has one root - the vertex is always at the root.

Question 17

What is the general form of a cubic equation?

Answer 17

ax^3 + bx^2 + cx + d

Question 18

How do you factorise a cubic when x is a factor (d = 0)?

Answer 18

Take x out so that you are left with a quadratic and factorise the quadratic.

Question 19

What is the Remainder Theorem?

Answer 19

For a polynomial f(x) the Remainder Theorem says:

• When you divide f(x) by (x - a) the remainder is f(a).–
• When you divide f(x) by (ax - b) the remainder is f(b/x).

Question 20

What is the factor theorem?

Answer 20

If you get a remainder of 0 when you divide f(x) by (x - a), then (x-a) must be a factor of f(x):
-If f(x) is a polynomial, and f(a) = 0, then (x - a) is a factor of f(x).

Question 21

How do you factorise a cubic when x isn’t a factor?

Answer 21

• Use the factor theorem (and a bit of trial and error) to find one of the factors of the cubic.
• Use your factor to find a quadratic that gives the cubic when multiplied by your factor.
• Factorise the quadratic you’ve found.

Question 22

How do you do algebraic long division?

Answer 22

• Divide the first term of the cubic by the first term of the divisor. Write this as the answer above the cubic
• Multiply the divisor by this under the first two terms of the cubic.
• Subtract this from the cubic terms above to get the new expression, bringing down the next term.
• Repeat this process until you are left with a constant.