The roots of the quadratic equation: algebraic and geometric meaning

In algebra, a square equation is a second-order equation. By the equation is meant a mathematical expression that has one or more unknowns in its composition. A second-order equation is a mathematical equation that has at least one square in the unknown degree. The quadratic equation is of the second order, the equation reduced to the form of an identity equal to zero. Solve the quadratic equation means the same thing as determining the roots of the quadratic equation. A typical quadratic equation in the general form:

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

Where W, T are the coefficients of the roots of the quadratic equation;

O is the free coefficient;

C is the root of the quadratic equation (always has two values of c1 and c2).

As already mentioned, the problem of solving the quadratic equation is finding the roots of the quadratic equation. In order to find them, it is necessary to find the discriminant:

N = T ^ 2 - 4 * W * O

A discriminant is needed to solve the formula for finding the root c1 and c2:

C1 = (-T + √N) / 2 * W and c2 = (-T - √N) / 2 * W

If in a quadratic equation of general form the coefficient at the root of T has a multiple value, then the equation is replaced by:

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

And its roots look like an expression:

C1 = [-U + √ (U ^ 2-W * O)] / W and c2 = [-U - √ (U ^ 2-W * O)] / W

Often the equation may have a slightly different form, when c_2 may not have a coefficient W. In this case, the above equation has the form:

C ^ 2 + F * c + L = 0

Where F is the coefficient at the root;

L is the free coefficient;

C is the root of the quadratic equation (always has two values of c1 and c2).

This kind of equation is called the reduced equation. The name "reduced" went from the reduction formula of a typical quadratic equation, if the coefficient at the root of W is 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 an even value of the coefficient at the root of F, the roots will have a solution:

C1 = -F + √ (F ^ 2-L) c2 = -F-√ (F ^ 2-L)

If we talk about quadratic equations, then we must remember the Vieta Theorem. It says that for the reduced quadratic equation, the following regularities exist:

C ^ 2 + F * c + L = 0

C1 + c2 = -F and c1 * c2 = L

In the general quadratic equation, the roots of the quadratic equation are related by the dependencies:

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

C1 + c2 = -T / W and c1 * c2 = O / W

Now let us consider the possible variants of quadratic equations and their solutions. There may be two of them, since if there is no member c_2, then the equation will no longer be square. Consequently:

1. W * c ^ 2 + T * c = 0 Variant of the quadratic equation without a free coefficient (term).

The solution is:

W * c ^ 2 = -T * c

C1 = 0, c2 = -T / W

2. W * c ^ 2 + O = 0 The variant of the quadratic equation without the second term, when the roots of the quadratic equation are equal in absolute value.

The solution is:

W * c ^ 2 = -O

C1 = √ (-O / W), c2 = - √ (-O / W)

All this was algebra. Consider the geometric meaning that the quadratic equation has. The second-order equation in geometry describes the parabola function. For high school students, the problem is often how to find the roots of the quadratic equation? These roots of the equation give a notion of how the graph of the function (parabola) intersects with the axis of coordinates - abscissas. If, solving the quadratic equation, we get an irrational solution of the roots, then there will be no intersection. If the root has one physical value, then the function crosses the abscissa axis in one place. If two roots, then, respectively, - two points of intersection.

It should be noted that an irrational root means a negative value under the root, when finding roots. The physical meaning is any positive or negative value. If only one root is found, the roots are assumed to be the same. The orientation of the curve on the Cartesian coordinate system can also be determined by the coefficients of the roots of W and T. If W has a positive value, then both branches of the parabola have an upward direction. If W has a negative value, then - down. Also, if the coefficient B has a positive sign, while W is also positive, then the vertex of the parabola function lies within the "y" from "-" infinity to "+" infinity, "c" from minus infinity to zero. If T is a positive value, and W is a negative value, then on the other side of the abscissa axis.