Quadratic Equations – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Quadratic Equations chapter of Mathematic. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Standard Form of a Quadratic Equation: ax² + bx + c = 0
Here, a is the coefficient of x², b is the coefficient of x, and c is the constant. This is the fundamental form of a quadratic equation, used for either graphical representation or solution finding.
Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
This formula gives solutions for any quadratic equation in standard form. The discriminant (b² - 4ac) indicates the nature of roots: real and distinct, real and equal, or complex.
Factored Form: a(x - r₁)(x - r₂) = 0
Where r₁ and r₂ are the roots of the equation. Useful for quickly determining roots when given a product of factors.
Sum and Product of Roots: r₁ + r₂ = -b/a, r₁r₂ = c/a
These equalities relate the roots of the quadratic equations to coefficients. They simplify finding roots without full factorization.
Vertex Form: y = a(x - h)² + k
In this form, (h, k) is the vertex of the parabola represented by the quadratic equation. Useful for graphing and understanding the graph's maximum/minimum points.
Discriminant: D = b² - 4ac
D is used to determine the nature of the roots. If D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are complex.
Completing the Square: ax² + bx = k → (x + b/(2a))² = (b² - 4ac)/(4a)
This method transforms a quadratic into vertex form. It’s useful for deriving the quadratic formula and understanding the parabola.
Roots of Unity: x² - (r₁ + r₂)x + r₁r₂ = 0
This formulation shows how the sum and product of the roots relate to the coefficients, reaffirming connections between algebra and geometry.
Graph of a Quadratic: y = ax² + bx + c
The graph is a parabola, opening upwards (a > 0) or downwards (a < 0). Understanding this helps in sketching quadratic functions and analyzing their behavior.
Quadratic Inequality: ax² + bx + c > 0
This is used to find the intervals where a quadratic is positive/negative. It involves determining the roots and testing intervals.
Equations
General Form: 2x² + x - 300 = 0
This particular equation is derived from a real-world scenario and can be solved using various methods, demonstrating practical applications of quadratics.
Example Quadratic Function: f(x) = x² - 5x + 6
This function can be analyzed to find its roots, vertex, and axis of symmetry, demonstrating the characteristics of its graph.
Factoring Example: x² - 7x + 10 = (x - 2)(x - 5) = 0
This shows how to factor a simple quadratic equation. Roots can be quickly identified as x = 2 or x = 5.
Graphical Representation: y = 2(x - 1)(x - 3)
Illustrates how to represent a quadratic equation in a factored manner, showing its roots clearly on a graph.
Inequality Example: x² - 4 < 0
This quadratic inequality can be solved to find intervals of x that satisfy the condition, enhancing critical thinking and problem-solving skills.
Using the Quadratic Formula: x = 4/3, -75/2 for 6x² + 5x + 4 = 0
An example using the quadratic formula to find non-integer solutions for a specific quadratic equation.
Area-Related Quadratic: x(x + 2) - 48 = 0
This equation arises from a real-world problem involving area dimensions, allowing for practical application during problem-solving.
Completing the Square: x² - 4x + 4 = 0 → (x - 2)² = 0
Shows how to transform and solve a quadratic equation by finding perfect square trinomials.
Vertex Calculation: V = (h, k) where h = -b/(2a), k = f(h)
Used to determine the vertex of the parabola, which assists in understanding its maximum/minimum value.
Real-life Application: x² + 8x + 16 = 0 → (x + 4)² = 0
Models a situation where the solution represents important dimensions or values in a context, reiterating the significance of quadratics.
