The roots of a quadratic equation: algebraic and geometric meaning

In algebra, the square is called the second-order equation.By equation imply a mathematical expression that has in its composition one or more unknown.The equation of the second order - a mathematical equation, which has at least one degree unknown in the square.Quadratic equation - second order equation shown to the form of the identity of zero.Solve the equation the square is the same that determine the square roots of the equation.Typical quadratic equation in the general form:

W * c ^ 2 + T * c + O = 0

where W, T - coefficients of the roots of a quadratic equation;

O - free coefficient;

c - the root of the quadratic equation (always has two values ​​c1 and c2).

As already mentioned, the problem of solving a quadratic equation - finding the roots of a quadratic equation.To find them, you need to find a discriminant:

N = T ^ 2 - 4 * W * O

discriminant formula needs to address the root finding c1 and c2:

c1 = (-T + √N) / 2 *W and c2 = (-T - √N) / 2 * W

If a quadratic equation of the general form factor at the root of T has a multiple of the value equation is replaced by:

W * c ^ 2 +2 * U * c +O = 0

and its roots look like the expression:

c1 = [-U + √ (U ^ 2-W * O)] / W and c2 = [-U - √ (U ^ 2-W * O)] / W

part of the equation may have a slightly different look when C_2 may not have the factor W. In this case, the above equation is:

c ^ 2 + F * c + L = 0

where F - the coefficient of the root;

L - free rate;

c - square root of (always has two values ​​c1 and c2).

This kind of equation is called a quadratic equation given.The name "given" came from the reduction formulas typical of a quadratic equation, if the ratio is at the root of W has a value of one.In this case the roots of the quadratic equation:

c1 = -F / 2 + √ [(F / 2) ^ 2-L)] and c2 = -F / 2 - √ [(F / 2) ^ 2-L)]

In the case of even values ​​of F at the root of the roots will have a solution:

c1 = -F + √ (F ^ 2-L) c2 = -F - √ (F ^ 2-L)

If we talk aboutquadratic equations, it is necessary to recall the Vieta theorem.It states that the above quadratic equation are the following laws:

c ^ 2 + F * c + L = 0

c1 + c2 = -F and c1 * c2 = L

In general quadratic equation roots of a quadratic equation are related dependencies:

W * c ^ 2 + T * c + O = 0

c1 + c2 = -T / W and c1 * c2 = O / W

Now consider the possible variants of quadratic equations and their solutions.In total there may be two, as if there will be no member c_2, then the equation will not be square.Therefore:

1. W * c ^ 2 + T * c = 0 Option quadratic equation without a constant coefficient (member).

The solution is:

W * c ^ 2 = -T * c

c1 = 0, c2 = -T / W

2. W * c ^ 2 + O = 0 Option quadratic equation without second term whensame modulo the roots of a quadratic equation.

The solution is:

W * c ^ 2 = -O

c1 = √ (-O / W), c2 = - √ (-O / W)

All this was algebra.Consider the geometric meaning of which has a quadratic equation.Second-order equations in the geometry described by a function of a parabola.For high school students often the task is to find the roots of a quadratic equation?These roots give an idea how to intersect the graph of the function (parabola) with the axis of coordinates - the abscissa.When deciding quadratic equation, we get the irrational decision of the roots, the crossing will not be.If the root has one physical value, the function intersects the x-axis at one point.If the two roots is respectively - the two points of intersection.

worth noting that under the irrational roots imply a negative value under the radical, in finding the roots.The physical value - any positive or negative value.In the case of finding only one root mean that the roots of the same.The orientation of the curve in the Cartesian coordinate system can also be pre-determined by factors at the root of W and T. If W has a positive value, then the two branches of the parabola are directed upwards.If W has a negative value, - downwards.Also, if the coefficient B has a positive sign, wherein W is also positive, the vertex of the parabola function is within the "y" from "-" to infinity "+" infinity, "c" in the range of minus infinity to zero.If T - positive value, and W - is negative, on the other side of the axis of abscissa.