# 1.4. Precalculus: Quadratics and polynomials

### Learning Objectives

You should be able to

• solve some quadratics by factorizing them
• use the discriminant to determine how many solutions a quadratic equation has, or how many roots a quadratic function has
• use the quadratic formula to find roots of a quadratic function or solutions to a quadratic equation
• re-write a quadratic by completing the square and use this to find roots and to find the vertex of the graph of the quadratic
• factorise some polynomials by using the factor theorem
• Use factorizes forms of polynomials to find their roots and on which regions they are positive or negative

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.
Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 1.4.1. Solving quadratics by factorization

$$x+a)(x+b) = x^2+(a+b)x+ab$$
$$c(x-d)(x-e) =0 \implies x=d \text{ or } x=e$$

#### 1.4.2. The quadratic formula and discriminant

$$ax^2+bx+c = 0 \iff x = \frac{-b\pm \sqrt{b^2-4ac} }{2a}$$      $$\Delta = b^2-4ac$$,
• $$\Delta>0 \implies 2\text{ distinct roots}$$,
• $$\Delta=0 \implies 1\text{ repeated root}$$,
• $$\Delta<0 \implies 0\text{ real roots}$$

#### 1.4.3. Completing the square

$$ax^2 + bx+c = a(x-x_0)^2 + y_0$$ has extreme at $$(x_0,y_0)$$
$$x^2+bx = \left(x+\frac{b}{2}\right)^2 - \frac{b^2}{4}$$

#### 1.4.4. Factorizing polynomials

Factor Theorem: If $$f(x)$$ is a polynomial and $$f(c)=0$$, then $$(x-c)$$ is a factor of $$f(x)$$.
Difference of two squares: $$a^2-b^2 = (a-b)(a+b)$$

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to. The additional resources gives links to external sources of information and some of these may also have further exercises to try.