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Mathematics
ALGEBRA · 01

Factorization & the quadratic formula

Expanding multiplies brackets into a sum. Factorizing reverses that process. We will trace every product, match the coefficients, then build a formula that solves any quadratic equation over the real or complex numbers.

01Where does every term come from?

The exponent 2 means two identical brackets, not squaring each term separately. Choose a formula below and play all four selections.

(a+b)2(a+b)^2
BRACKET 1( a + b )BRACKET 2( a + b )a²ababb²
Selection 1a⋅a=a2a\cdot a=a^2
Selection 2a⋅b=aba\cdot b=ab
Selection 3b⋅a=abb\cdot a=ab
Selection 4b⋅b=b2b\cdot b=b^2
Multiplication distributes: choose one term from EACH bracket, multiply, and add all four possible products.
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(a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2
x2+6x+9=x2+2⋅3⋅x+32=(x+3)2x^2+6x+9=x^2+2\cdot3\cdot x+3^2=(x+3)^2

Read the identity backwards: x takes the role of a and 3 takes the role of b.

(a±b)2=a2±2ab+b2(a\pm b)^2=a^2\pm2ab+b^2

Use the matching sign throughout. The last term is always +b², because both b·b and (−b)(−b) are b². In contrast, (a+b)(a−b) has a last term −b².

02Use the expansion to find the factors.

For ax²+bx+c, suppose the factors are (Ax+B)(Cx+D). Capital A, B, C, D are unknown constants, distinct from the original coefficients a, b, c. Expanding gives the three conditions below.

(Ax+B)(Cx+D)=ACx2+(AD+BC)x+BD(Ax+B)(Cx+D)=ACx^2+(AD+BC)x+BD
AC=a,AD+BC=b,BD=cAC=a,\qquad AD+BC=b,\qquad BD=c
2x2−2x−42x^2-2x-4
x coefficientsconstants1x-22x2AD = 2BC = -4AD + BC = -2AC = 2BD = -4
(x−2)(2x+2)(x-2)(2x+2)

Choose factors for the leading coefficient and the constant. The crossing products must add to the middle coefficient. The display always shows the product you actually chose.

Matched: -2 is the required coefficient -2. The factorization is valid.
2x2−2x−4=(x−2)(2x+2)2x^2-2x-4=(x-2)(2x+2)1⋅2=2,(−2)⋅2=−4,1⋅2+(−2)⋅2=−21\cdot2=2,\quad(-2)\cdot2=-4,\quad1\cdot2+(-2)\cdot2=-2

If this expression equals zero, at least one factor must be zero: x=2 or x=−1. Integer factors are convenient, but not every quadratic has them. Completing the square is the general method.

03Build the quadratic formula, one equality at a time.

A quadratic equation has the standard form ax²+bx+c=0, with a≠0. The goal is to isolate x by turning the left side into a square. Click any line to inspect that step.

Area example: x² + 6x. Split 6x into two strips of area 3x, and move one around the square. Here x > 0.

x²3x3x3²x3x3
x2+6x=(x+3)2−9x^2+6x=(x+3)^2-9
The missing corner is 3 × 3 = 9. Adding it completes a square of side x+3. For negative b, use the same algebra; do not interpret a negative number as a physical side length.
DERIVATION 1 / 9
ax2+bx+c=0(a≠0)ax^2+bx+c=0\quad(a\ne0)

The leading coefficient a is nonzero; otherwise this is not quadratic.

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What does the square root tell us?

Δ=b2−4ac=0\Delta=b^2-4ac=0

Zero Δ: the two signs give the same root, a repeated real root.

x=−3x=-3