**1.4.1. Solving quadratics by factorization**

\(x+a)(x+b) = x^2+(a+b)x+ab\)

\(c(x-d)(x-e) =0 \implies x=d \text{ or } x=e\)

You should be able to

- recognize a quadratic function
- 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

Our recommendation is to:

**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.**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.**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.**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,

\(x+a)(x+b) = x^2+(a+b)x+ab\)

\(c(x-d)(x-e) =0 \implies x=d \text{ or } x=e\)

- \( \Delta>0 \implies 2\text{ distinct roots}\),
- \( \Delta=0 \implies 1\text{ repeated root}\),
- \( \Delta<0 \implies 0\text{ real roots} \)

\(x^2+bx = \left(x+\frac{b}{2}\right)^2 - \frac{b^2}{4} \)

**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:**
\(a^2-b^2 = (a-b)(a+b)\)

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.