#A04 · Algebra & Number Theory · Lv.2View in 3D Mind-Map

Quadratic Formula & Discriminant

Brahmagupta & Al-Khwarizmi · 628 CE

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

ax^2 + bx + c = 0 \quad (D = b^2 - 4ac)

Solves any quadratic equation directly from its coefficients via completing the square, while the discriminant D = b² - 4ac classifies real, repeated, or complex roots.

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Coeff a

Coeff b

Constant c

D = (-3)^2 - 4(1)(-4) = 25.00

Roots x₁, x₂

4.00, -1.00

Discriminant D

25.00 (실근 2)

Vertex

(1.50, -6.25)

Prerequisite Path (4 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

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

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

🚀 Next Unlocks (Derived Formulas) (5)

#A05Vieta's Formulas (Roots & Coefficients)#A09Complex Numbers & Polar Plane #G07Conic Sections (Parabola, Ellipse, Hyperbola)#C10Differential Equations & Harmonic Oscillator #L08Eigenvalues & Eigenvectors (Av = λv)

Geometric & Intuitive Insight

이차함수 포물선 y = ax² + bx + c 의 대칭축 x = -b/(2a) 로부터 좌우로 ±√(b²-4ac)/(2a) 만큼 떨어진 두 점이 바로 x축과 만나는 두 근입니다.

Step-by-Step Derivation

Step 1.1단계: ax² + bx + c = 0 의 양변을 a로 나눕니다 → x² + (b/a)x = -c/a
Step 2.2단계: 완전제곱식을 만들기 위해 양변에 (b/2a)² 을 더합니다 → (x + b/2a)² = (b² - 4ac) / 4a²
Step 3.3단계: 제곱근을 취하고 x에 대해 정리하면 x = (-b ± √(b² - 4ac)) / 2a 가 유도됩니다.

AI & Coding Real-World Use

3D Ray-Sphere intersection in ray tracing shaders and projectile trajectory impact time calculation

#3D Ray Tracing#Physics Collision#Optimization Parabola

const D = b*b - 4*a*c; const x1 = (-b + Math.sqrt(D)) / (2*a);

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