LESSON 23

QUADRATIC EQUATIONS

The Mathematics of Curves and Parabolas

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2 — the highest exponent of the variable is 2. The standard form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a U-shaped curve called a parabola. The solutions to a quadratic equation are called roots or zeros — the x-values where the parabola crosses the x-axis.

Quadratic equations appear throughout science, engineering, and everyday life. They describe the path of a thrown ball, the shape of satellite dishes and car headlights, the profit curves of businesses, and the behavior of electrical circuits. There are three main methods for solving them: factoring, completing the square, and the quadratic formula.

Solving by Factoring

If a quadratic expression can be factored, set each factor equal to zero and solve. This method uses the Zero Product Property: if A × B = 0, then A = 0 or B = 0. First, write the equation in standard form (ax² + bx + c = 0). Then factor the left side. Finally, set each factor equal to zero and solve each mini-equation. Always check your solutions by substituting back into the original equation.

Example:x² + 5x + 6 = 0 Factor: (x + 2)(x + 3) = 0 Set each factor to zero: x + 2 = 0 → x = −2 x + 3 = 0 → x = −3 Solutions: x = −2 and x = −3

The Quadratic Formula

The quadratic formula solves any quadratic equation ax² + bx + c = 0, even when factoring is difficult or impossible. The formula is x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant. If the discriminant is positive, there are two real solutions. If it equals zero, there is exactly one real solution (a repeated root). If it is negative, there are no real solutions — the parabola does not cross the x-axis.

Example:2x² − 4x − 6 = 0 (a=2, b=−4, c=−6) Discriminant: (−4)² − 4(2)(−6) = 16 + 48 = 64 x = (4 ± √64) / 4 = (4 ± 8) / 4 x = 12/4 = 3 or x = −4/4 = −1 Solutions: x = 3 and x = −1

Completing the Square

Completing the square rewrites a quadratic in vertex form: a(x − h)² + k = 0, where (h, k) is the vertex of the parabola. Step 1: move the constant to the right side. Step 2: divide all terms by a if a ≠ 1. Step 3: add (b/2)² to both sides to create a perfect square trinomial on the left. Step 4: factor the left side and solve by taking the square root of both sides. This method also reveals the vertex of the parabola.

Example:x² + 6x − 7 = 0 x² + 6x = 7 Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16 (x + 3)² = 16 x + 3 = ±4 x = 1 or x = −7

Parabolas and the Vertex

The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upward (minimum point). If a < 0, it opens downward (maximum point). The vertex is the highest or lowest point, found at x = −b / 2a. The axis of symmetry is the vertical line through the vertex. The y-intercept is the point (0, c). Understanding the vertex and direction of opening lets you sketch the parabola and interpret its meaning in real-world problems.

Example:y = x² − 4x + 3 Vertex x: x = −(−4) / 2(1) = 2 Vertex y: y = (2)² − 4(2) + 3 = −1 Vertex: (2, −1) — minimum point Opens upward (a = 1 > 0) y-intercept: (0, 3)

Key Facts

The quadratic formula was known in various forms to ancient Babylonians, Greeks, Indians, and Arabs. The modern symbolic version using ± and the square root sign was standardized in Europe during the 16th and 17th centuries.

Every parabolic satellite dish, car headlight reflector, and telescope mirror uses the geometric property that all rays parallel to the axis of a parabola reflect through a single focal point — a direct consequence of the quadratic equation.

Galileo proved in 1638 that projectiles follow parabolic paths (ignoring air resistance). This means the trajectory of a basketball, a cannonball, or a water fountain jet is governed by a quadratic equation.

The discriminant b² − 4ac tells you everything about the nature of the roots before you solve. A perfect square discriminant means rational roots; a non-perfect-square positive discriminant means irrational roots; zero means one repeated root; negative means complex roots.

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