Practice

Factoring quadratics

Rewriting quadratic trinomials and quadratic equations as a product of factors instead of a sum of terms. Lets you solve equations quickly, simplify messy expressions, and spot the roots immediately.

Connections:

Completing the square is required to factor a quadratic trinomial in the general case, so you absolutely need to know how to use this method.
Adding and compensating
Factoring a quadratic trinomial uses the discriminant and the general root formula for quadratic equations. You need all of that, no excuses.
The quadratic formula
Discriminant
Biquadratic equation

Statistics:

  • Statement1
  • Important2
  • Problem12
  • Updated
Article
Summary
Practice

No-brainer roots

😀
Elementary
Solve the equation:
(x+6)(x−10)=0\left(x + 6\right)\left(x - 10\right) = 0

Factoring signs

Method
😀
Elementary
Without finding the roots, determine what signs the numbers t and k will have when the trinomials are factored:
1) x2+80x+31) \ x^2 + 80x + 3
2) y2−2y+18 2) \ y^2 - 2y + 18
3) z2−5z−14 3) \ z^2 - 5z - 14
4) w2+4w−5 4) \ w^2 + 4w - 5

There and back again

😀
Elementary
Factor the first expression, and rewrite the expression on the right in standard quadratic trinomial form:
1)x2−2x−31) \enspace x^2 - 2x - 3
2)(x+5)(x+1) 2) \enspace (x + 5)(x + 1)

Factoring

😀
Elementary
Factor the quadratic trinomial and find the roots of the corresponding quadratic equation:
x2−6x−1x^2 - 6x - 1

Equation architect

Method
😀
Elementary
Build a quadratic equation with the given roots.
Roots 2 and 3. Coefficient A = 1.

One root is one

Method
Elegant
🤔
Intermediate
If one of the roots of a quadratic equation is equal to 1, how are its coefficients related? What is the second root in that case?

Bracket chaos

🤔
Intermediate
Simplify the equation down to a product of factors and find the roots:
(x+5)(x−1)=3x+7(x + 5)(x - 1) = 3x + 7

Master of both elements

Elegant
🤔
Intermediate
Get really damn good at handling quadratic trinomials that are written in two forms at once: as a sum and as a product of factors, yin and yang style. Simplify the expression:
(x−3)(x+5)−x2+9(x - 3)(x + 5) - x^2 + 9

Factoring biquadratic trinomials

🤯
Advanced
Factor the biquadratic trinomial as far as possible:
x4−5x2+4x^4 - 5x^2 + 4

Exploring the trinomial plane

🤯
Advanced
Let’s explore the coordinate plane of “rectangular” quadratic trinomials together:
Study what kinds of quadratic trinomials appear at the locations marked with red dots. Describe the general form and give a couple of examples with specific numbers.
Factoring quadratics