Quadratic Equations: Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• In elementary algebra, those symbols represent quantities without fixed values, known as variables.

Quadratic Equations

• A quadratic equation is a polynomial equation of the second degree.
• The general form is ax² + bx + c = 0, where a ≠ 0.
• a, b, and c are coefficients, which can be numbers or symbolic expressions.

Forms of Quadratic Equations

• Standard Form: ax² + bx + c = 0
• Factored Form: a(x - r₁)(x - r₂) = 0, where r₁ and r₂ are the roots of the equation.
• Vertex Form: a(x - h)² + k = 0, where (h, k) is the vertex of the parabola represented by the equation.

Solving Quadratic Equations

• Several methods exist to find the values of x that satisfy the quadratic equation.

Factoring

• Method involves rewriting the quadratic expression as a product of two linear factors.
• Applicable when the quadratic expression can be easily factored.
• Set each factor equal to zero and solve for x to find the roots.
• Example: x² - 5x + 6 = (x - 2)(x - 3) = 0, so x = 2 or x = 3.

Completing the Square

• Method involves transforming the quadratic equation into a perfect square trinomial.
• Useful when the quadratic equation cannot be easily factored.
• Steps:
  • Divide the equation by "a" if a ≠ 1.
  • Move the constant term (c/a) to the right side of the equation.
  • Add (b/2a)² to both sides of the equation to complete the square.
  • Express the left side as a squared binomial.
  • Take the square root of both sides and solve for x.
• Example: x² + 6x + 5 = 0 => x² + 6x = -5 => x² + 6x + 9 = -5 + 9 => (x + 3)² = 4 => x + 3 = ±2 => x = -1 or x = -5.

Quadratic Formula

• A general formula that provides the solutions to any quadratic equation.
• Solutions are: x = (-b ± √(b² - 4ac)) / 2a
• Derived by completing the square on the standard form of the quadratic equation.
• Use when factoring or completing the square is not straightforward.

Discriminant

• The discriminant (Δ) is the expression b² - 4ac within the quadratic formula.
• Determines the nature of the roots of the quadratic equation:
  • If Δ > 0: Two distinct real roots.
  • If Δ = 0: One real root (a repeated root).
  • If Δ < 0: Two complex conjugate roots.

Nature of Roots

• Real Roots: Roots that are real numbers, can be rational or irrational.
• Rational Roots: Roots that can be expressed as a ratio of two integers. Occur when the discriminant is a perfect square.
• Irrational Roots: Roots that cannot be expressed as a ratio of two integers. Occur when the discriminant is positive but not a perfect square.
• Complex Roots: Roots that involve the imaginary unit 'i' (where i² = -1). Occur when the discriminant is negative. Complex roots always occur in conjugate pairs (a + bi and a - bi).

Vieta's Formulas

• Relate the coefficients of a polynomial to sums and products of its roots.
• For a quadratic equation ax² + bx + c = 0 with roots r₁ and r₂:
  • Sum of roots: r₁ + r₂ = -b/a
  • Product of roots: r₁ * r₂ = c/a
• Useful for finding the sum and product of the roots without explicitly solving the equation.
• Can be used to construct a quadratic equation given its roots.

Applications of Quadratic Equations

• Physics: Projectile motion, where the height of a projectile is modeled by a quadratic equation.
• Engineering: Designing parabolic structures like bridges and satellite dishes.
• Economics: Modeling cost, revenue, and profit functions.
• Computer Graphics: Creating curves and surfaces.
• Optimization Problems: Maximize or minimize quantities subject to constraints.

Graphing Quadratic Equations

• The graph of a quadratic equation y = ax² + bx + c is a parabola.
• The vertex of the parabola is the point (h, k), where h = -b/2a and k = f(h).
• The axis of symmetry is the vertical line x = h, which passes through the vertex.
• The parabola opens upwards if a > 0 and downwards if a < 0.
• The x-intercepts are the real roots of the equation ax² + bx + c = 0.
• The y-intercept is the point (0, c).

Vertex of a Parabola

• The vertex represents the maximum or minimum point of the parabola.
• If a > 0, the vertex is the minimum point.
• If a < 0, the vertex is the maximum point.
• The x-coordinate of the vertex is given by h = -b/2a.
• The y-coordinate of the vertex is found by substituting h into the equation: k = f(h).

Axis of Symmetry

• The axis of symmetry is a vertical line that divides the parabola into two symmetric halves.
• Its equation is x = h, where h is the x-coordinate of the vertex.
• The parabola is symmetric about this line.

X-Intercepts

• The x-intercepts are the points where the parabola intersects the x-axis.
• These points correspond to the real roots of the quadratic equation.
• To find the x-intercepts, set y = 0 and solve for x.
• The number of x-intercepts depends on the discriminant:
  • Two x-intercepts if Δ > 0.
  • One x-intercept if Δ = 0.
  • No x-intercepts if Δ < 0.

Y-Intercept

• The y-intercept is the point where the parabola intersects the y-axis.
• To find the y-intercept, set x = 0 and solve for y.
• The y-intercept is always the point (0, c).

Quadratic Inequalities

• Involve comparing a quadratic expression to a constant or another expression using inequality signs (<, >, ≤, ≥).
• To solve:
  • Rewrite the inequality so that one side is zero.
  • Find the roots of the corresponding quadratic equation.
  • Use the roots to divide the number line into intervals.
  • Test a value from each interval to determine where the inequality is satisfied.
  • Write the solution set as an interval or union of intervals.

Systems of Equations Involving Quadratics

• Solving systems of equations where at least one equation is quadratic.
• Methods:
  • Substitution: Solve one equation for one variable and substitute into the other equation.
  • Elimination: Manipulate the equations to eliminate one variable.
• Solutions represent the points of intersection of the curves represented by the equations. A line and a parabola can intersect at two points, one point, or no points. Two parabolas can intersect at up to four points.