Math FoundationsEquationsQuadratic Equations

Quadratic formula

A clear derivation of the quadratic formula with a detailed explanation of every step. Learn what the discriminant is, where it comes from, and how it tells you how many roots a quadratic equation has.
Key elements:
Connections:
The quadratic formula is derived by completing the square, so you really need to know how that works!
Adding and compensating
Statistics:
Term2
Statement1
Important2
Problem14
Article
Summary
Practice

Quadratic equations

😀
Elementary
Solve the quadratic equation using the quadratic formula:
j2−4j−60=0j^2 - 4j - 60 = 0

Biquadratic equations

😀
Elementary
Solve the biquadratic equation:
j4−20j2+64=0j^4 - 20j^2 + 64 = 0

David and Goliath

😀
Elementary
360j2−660j−2100=0360j^2 - 660j - 2100 = 0

Even coefficient

Method
🤔
Intermediate
Derive a general root formula for a quadratic equation assuming the coefficient B is an even number. Also derive a separate formula for the special case where B is even and A = 1.

Roots with letters

🤔
Intermediate
Derive expressions for the roots of the equations:
x2−7ax+12a2=0x^2 - 7ax + 12a^2 = 0

Sridhara's method

Elegant
🤔
Intermediate
The Indian mathematician Sridhara came up with a way to derive the quadratic formula back in the 8th century, in a method where fractions don't appear until the very last step. His first move is to multiply both sides by 4A. Finish the derivation.

Biquadratic roots formula

🤔
Intermediate
Derive the general root formula for a biquadratic equation:
Ax4+Bx2+C=0Ax^4 + Bx^2 + C = 0

Substitution method

🤯
Advanced
Another way to derive the quadratic formula is to make a clever change of variables so that you get the simplest incomplete quadratic equation of the form u2=su^2 = s, which is solved instantly. Use this substitution:
x=u−B2Ax = u - \frac{B}{2A}
Carry out the substitution and finish the derivation of the quadratic formula.

Sneaky discriminant

Elegant
🤯
Advanced
Solve the equation:
x2−(2+1)x+2=0x^2 - (\sqrt{2} + 1)x + \sqrt{2} = 0

Trinomial equations

Method
🤯
Advanced
A trinomial equation is any equation that can be reduced to the form:
Af2(x)+Bf(x)+C=0Af^2(x) + Bf(x) + C = 0
Figure out how to solve equations like this, and try it on:
2x6−3x3+1=02x^6 - 3x^3 + 1 = 0

Reciprocal equations

Method
Elegant
🤯
Advanced
Reciprocal equations, also called palindromic or symmetric equations, are equations where the coefficients mirror each other around the equation's “center”:
Figure out how to solve such equations and try it on the one above.
Quadratic formula