Properties and methods for finding the roots of a quadratic equation

The world is arranged in such a way that the solution of a large number of problems is reduced to finding the roots of the quadratic equation. The roots of the equations are important for describing various laws. This was known to the surveyors of ancient Babylon. Astronomers and engineers were also forced to solve such problems. As far back as the 6th century AD, the Indian scientist Ariabhata developed the foundations for finding the roots of the quadratic equation. The formulas acquired a finished look in the 19th century.

General concepts

We suggest that you familiarize yourself with the basic laws of quadratic equalities. In general, equality can be written as follows:

ax ² + bx + c = 0,

The number of roots of the quadratic equation can be equal to one or two. A quick analysis can be done using the concept of discriminants:

D = b ² - 4ac

Depending on the calculated value, we get:

• For D> 0, there are two different roots. The general formula for determining the roots of the quadratic equation looks like (-b ± √D) / (2a).
• D = 0, in this case the root is one and corresponds to the value x = -b / (2a)
• D <0, for a negative value of the discriminant, there is no solution to the equation.

Note: if the discriminant is negative, the equation has no roots only in the region of real numbers. If the algebra is expanded to the concept of complex roots, then the equation has a solution.

We give a chain of actions confirming the formula for finding the roots.

From the general form of the equation, it follows:

ax ² + bx = -c

We multiply the right and left sides by 4a and add b ² , we get

4a ² x ² + 4abx + b ² = -4ac + b ²

Transform the left side into the square of the polynomial (2ax + b) ² . We extract the square root from both sides of the equation 2ax + b = -b ± √ (-4ac + b ² ), transfer the coefficient b to the right side, we get:

2ax = -b ± √ (-4ac + b ² )

This implies:

x = (-b ± √ (b ² - 4ac))

Which was required to show.

Special case

In some cases, the solution to the problem can be simplified. So, with an even coefficient b we get a simpler formula.

Denote k = 1 / 2b, then the general formula of the roots of the quadratic equation takes the form:

x = (-k ± √ (k ² - ac)) / a

For D = 0, we obtain x = -k / a

Another particular case is the solution of the equation for a = 1.

For the form x ² + bx + c = 0, the roots will be x = -k ± √ (k ² - c) with a discriminant greater than 0. For the case when D = 0, the root will be determined by a simple formula: x = -k.

Using Charts

Any person, without even suspecting it, is constantly faced with physical, chemical, biological and even social phenomena that are well described by a quadratic function.

Note: a curve constructed on the basis of a quadratic function is called a parabola.

Here are some examples.

1. When calculating the flight path of the projectile, the property of movement along the parabola of a body released at an angle to the horizon is used.
2. The property of parabolas to evenly distribute the load is widely used in architecture.

Understanding the importance of a parabolic function, we’ll figure out how to study its properties using a graph using the concepts of “discriminant” and “roots of a quadratic equation”.

Depending on the magnitude of the coefficients a and b, there are only six options for the position of the curve:

1. The discriminant is positive, a and b have different signs. The branches of the parabola look up, the quadratic equation has two solutions.
2. The discriminant and coefficient b are equal to zero, the coefficient a is greater than zero. The graph is located in the positive zone, the equation has 1 root.
3. The discriminant and all coefficients have positive values. The quadratic equation has no solution.
4. The discriminant and coefficient a are negative, b is greater than zero. The branches of the graph are directed downward, the equation has two roots.
5. The discriminant and the coefficient b are equal to zero, the coefficient a is negative. Parabola looks down, the equation has one root.
6. The values ​​of the discriminant and all coefficients are negative. There are no solutions, the function value is completely in the negative zone.

Note: the variant a = 0 is not considered, since in this case the parabola degenerates into a straight line.

All of the above is well illustrated by the figure below.

Examples of solving problems

Condition: using common properties, make a quadratic equation, whose roots are equal to each other.

Decision:

by the assumption of the problem x ₁ = x ₂ , or -b + √ (b ² - 4ac) / (2a) = -b + √ (b ² - 4ac) / (2a). Simplify the entry:

-b + √ (b ² - 4ac) / (2a) - (-b - √ (b ² - 4ac) / (2a)) = 0, open the brackets and give similar terms. The equation takes the form 2√ (b ² - 4ac) = 0. This statement is true when b ² - 4ac = 0, hence b ² = 4ac, then we substitute the value b = 2√ (ac) into the equation

ax ² + 2√ (ac) x + c = 0, in the reduced form we get x ² + 2√ (c / a) x + c = 0.

Answer:

for a not equal to 0 and any c there is only one solution if b = 2√ (c / a).

For all its simplicity, quadratic equations are of great importance in engineering calculations. Almost any physical process can be described with some approximation using power functions of order n. The quadratic equation will be the first such approximation.