Linear Equation vs Quadratic Equation
In mathematics, algebraic equations are equations which are formed using polynomials. When explicitly written the equations will be of the form P(x) = 0, where x is a vector of n unknown variables and P is a polynomial. For example, P(x,y) = x4 + y3 + x2y + 5=0 is an algebraic equation of two variables written explicitly. Also, (x+y)3=3x2y – 3zy4 is an algebraic equation, but in implicit form. It will take the form Q(x,y,z) = x3 + y3 + 3xy2+3zy4= 0, once written explicitly.
An important characteristic of an algebraic equation is its degree. It is defined to be the highest power of the terms occurring in the equation. If a term consists of two or more variables, the sum of the exponents of each variable will be taken to be the power of the term. Observe that according to this definition P(x,y) = 0 is of degree 4 while Q(x,y,z) = 0 is of degree 5.
Linear equations and quadratic equations are two different types of algebraic equations. The degree of the equation is the factor that differentiates them from the rest of the algebraic equations.
What is a linear equation?
A linear equation is an algebraic equation of degree 1. For example, 4x + 5 = 0 is a linear equation of one variable. x + y + 5z = 0 and 4x = 3w + 5y + 7z are linear equations of 3 and 4 variables respectively. In general, a linear equation of n variables will take the form m1x1 +m2x2 +…+ mn-1xn-1 + mnxn = b. Here, xi’s are the unknown variables, mi’s and b are real numbers where each of mi is non-zero.
Such an equation represents a hyper plane in the n-dimensional Euclidean space. In particular, a two variable linear equation represents a straight line in Cartesian plane and a three variable linear equation represents a plane on Euclidean 3-space.
What is a quadratic equation?
A quadratic equation is an algebraic equation of the second degree. x2 + 3x + 2 = 0 is a single variable quadratic equation. x2 + y2 + 3x= 4 and 4x2 + y2 + 2z2 + x + y + z = 4 are examples of quadratic equations of 2 and 3 variables respectively.
In the single variable case, the general form of a quadratic equation is ax2 + bx + c = 0. Where a, b, c are real numbers out of which ‘a’ is non-zero. The discriminant ∆ = (b2 – 4ac) determines the nature of the roots of the quadratic equation. The roots of the equation will be real distinct, real similar and complex according as ∆ is positive, zero and negative. The roots of the equation can be easily found using the formula x = (- b ± √∆ ) / 2a.
In the two variable case, the general form would be ax2 + by2 + cxy + dx + ex + f = 0, and this represents a conic (parabola, hyperbola or ellipse) in Cartesian plane. In higher dimensions, this type of equations represents hyper-surfaces known as quadrics.
What is the difference between linear and quadratic equations?
• A linear equation is an algebraic equation of degree 1, whereas a quadratic equation is an algebraic equation of degree 2.
• In the n-dimensional Euclidean space, the solution space of an n-variable linear equation is a hyper plane while that of an n-variable quadratic equation is a quadric surface.