Advanced Quadratics Theory

Question 1

How do you know when there are two repeated real roots for ax²+bx+c?

Answer 1

You find when the discriminant is 0.

So solve b²-4ac=0.

Question 2

How do you know when there are no real roots for ax²+bx+c?

Answer 2

You find when the discriminant less than 0.

So solve b²-4ac<0.

Question 3

How do you know when there are two real roots for ax²+bx+c?

Answer 3

You find when the discriminant greater than 0.

So solve b²-4ac>0.

Question 4

When you need to use the “complete the square” method?

Answer 4

Whenever you need the form of the equation to tell you the vertex.

f(x)=(x-h)²+k

Question 5

What’s the quadratic formula?

Answer 5

Question 6

What are the three ways to solve a quadratic equation?

Answer 6

• Factorization
• Completing the square
• The quadratic formula

Question 7

What are the steps to solving a quadratic equation by factorising?

Answer 7

1. Get everything on one side, leaving just 0 on the other. (i.e. x²-2x+1=0).
2. Factorise, so you get something like (x+a)(x+b)=0.
3. Solve each parenthesis for x. (i.e. x+a=0 and x+b=0)

Question 8

How can you recognize that an equation is quadratic?

Answer 8

Some variable is squared.

It can always be written as:

ax²+bx+c=0

Question 9

What are the steps for factorising a quadratic equation?

ax²+bx+c=0

Answer 9

1. Factor out any common factors. (So anything that goes into all of the terms. i.e. 6x²-2x-8 = 2 (3x²-x-4))
2. Make a list of all the factors that multiply to make ac.
3. Check to see which two factors from step two add to the b term.
4. Rewrite the equation, separating the middle term into the two factors you found.
5. Take the common factors out of the first two terms and the last two, and regroup.

For example, 6x²-2x+8:

1. 6x²-2x+8=2(3x²-x-4). Now I need to factorise 3x²-x-4
2. 3•-4=-12, so the factors are 1•-12, -1•12, 2•-6, -2•6, 3•-4, -3•4
3. Do any of the pairs add to -1? Yep, 3+-4=-1.
4. 3x²+3x-4x-4
5. 3x(x+1) - 4(x+1) = (3x-4)(x+1). So the answer is 6x²-2x+8=2(3x-4)(x+1).

Question 10

What does the graph of f(x)=x² look like? What are the vertex and line of symmetry?

Answer 10

Vertex: (0,0)

Line of symmetry: x=0

Question 11

How do you turn

f(x)=ax²+bx+c

into the “vertex” form

f(x)=a(x-h)²+k?

Answer 11

Question 12

What is a coefficient?

Answer 12

A coefficient is a number that is multiplying (it is attached to) a variable, like an x or any other letter.

e.g.

3x² + 2x - 4

3 is the coefficient of x²,

and 2 is the coefficient of x.

Question 13

What is the leading coefficient?

Answer 13

The leading coefficient, is the number that is attached to the x with the highest power.

e.g. in the trinomial

4x² + 3x + 9

the leading coefficinent is 4

Question 14

What is a constant?

Answer 14

A constant is a number that is not attached to a variable (like an x, or any other number).

e.g. 3x² + 4x + 7

7 is the constant.

3 and 4 are coefficients.

3 is the leading coefficient.

Question 15

What is a quadratic?

Answer 15

The general quadratic looks like:

ax² + bx + c

“Quad” means “square”, which in math means to the power of 2. So a quadratic is an equation that contains an x², and this is the highest power. There can’t be, for example x³, nor x⁴, and so on. Only the square is the highest power. Quadratic expression contains three terms, which makes it a trinomial.