Instruction

1

Write the quadratic equation in the form ax2 + bx + c = 0

Example:

Original equation: 12 + x2= 8x

Correct the equation: x2 - 8x + 12 = 0

Example:

Original equation: 12 + x2= 8x

Correct the equation: x2 - 8x + 12 = 0

2

Apply the vieta theorem, according to which the sum of the roots of the equation will be equal to the number "b", taken with the opposite sign, and their work - a "c".

Example:

In this equation b=-8, c=12, respectively:

x1+x2=8

x1∗x2=12

Example:

In this equation b=-8, c=12, respectively:

x1+x2=8

x1∗x2=12

3

Learn positive and negative numbers are roots of equations. If the product and the sum of the roots is positive number, each of the roots is positive. If the product of the roots is positive and the sum of the roots is negative, then both root – negative. If the product of the roots is negative, the roots one root is the " + " sign and another sign "-" In this case, you must use an additional rule: "If the sum of the roots is a positive number greater than the module root is also positive and if the sum of the roots is negative - the larger the module root is negative."

Example:

In this equation the sum and product of positive numbers: 8 and 12, then both the root - the positive number.

Example:

In this equation the sum and product of positive numbers: 8 and 12, then both the root - the positive number.

4

Solve this system of equations by choosing from the roots. Will be convenient to begin with the selection of the multipliers, and then, to check, substitute each pair of multipliers into the second equation and test whether the sum of these roots solution.

Example:

x1∗x2=12

Suitable pairs of roots are respectively: 12 and 1, 6 and 2, 4 and 3

Check out the pair using equation x1+x2=8. Pair

12 + 1 ≠ 8

6 + 2 = 8

4 + 3 ≠ 8

Accordingly, the roots of an equation are numbers 6 and 8.

Example:

x1∗x2=12

Suitable pairs of roots are respectively: 12 and 1, 6 and 2, 4 and 3

Check out the pair using equation x1+x2=8. Pair

12 + 1 ≠ 8

6 + 2 = 8

4 + 3 ≠ 8

Accordingly, the roots of an equation are numbers 6 and 8.

Note

In this example, was a variant of the quadratic equation where a=1. To the same way to solve a full quadratic equation, where a&ne 1, it is necessary to make an auxiliary equation, giving "a" to the unit.

Useful advice

Use this method for solving equations in order to quickly find the roots. Also it will help if you need to solve the equation in the mind, without resorting to the records.

# Advice 2: How to find the root of the equation

The equation is called equality of the form f(x,y,...)=g(x,y,..) where f and g are functions of one or several variables. To find the root of an equation means finding the set of arguments at which this equality occurs.

You will need

- Knowledge in mathematical analysis.

Instruction

1

Let's say you have an equation of the form: x+2=x/5. For starters, move all components of this equality from the right side to the left, changing the sign of the component on the opposite. In the right part of this equation will be zero, i.e., we get the following: x+2-x/5 = 0.

2

We give similar terms. Will receive the following: 4x/5 + 2 = 0.

3

Further, from the obtained equation, we find the unknown addend, which in this case is X. the resulting value of the unknown variable will be the solution to the original equation. In this case, we get the following: x = -2,5.

Note

The solutions may be extra roots. They will not be the solution to the original equation, even if you do all correctly decided. Be sure to check all of the solution.

Useful advice

The obtained values of the unknown always check. It's easy to do, substituting the obtained value in the original equation. If the equality is true, then the decision is correct.

# Advice 3: How to find the negative root of the equation

If the substitution number in the equation is true equality, such a number is called a root. The roots can be positive, negative or zero. Among the many roots of the equation distinguish a maximum and a minimum.

Instruction

1

Find all the roots of equations, among them select negative, if any. For example, the quadratic equation 2x2-3x+1=0. Apply the formula for finding roots of a quadratic equation: x(1,2)=[3±√(9-8)]/2=[3±√1]/2=[3±1]/2, then x1=2, x2=1. It is easy to see that the negative is not among them.

2

To find the roots of quadratic equations you can also use the theorem of vieta. According to this theorem x1+x1=-b, x1∙x2=c, where b and c are respectively the coefficients of the equation x2+bx+c=0. Using this theorem, you can calculate the discriminant b2-4ac, which in some cases can significantly simplify the task.

3

If in a square equation the coefficient of x is even, can be used not primary, and short formulas for finding roots. If the basic formula looks like x(1,2)=[-b±√(b2-4ac)]/2a, then in abbreviated form it is written as: x(1,2)=[-b/2±√(b2/4-ac)]/a. If in a square equation, no intercept, simply make x the brackets. And sometimes the left part of the folds to complete the square: x2+2x+1=(x+1)2.

4

Are the types of equations that do not give a single number, a whole lot of solutions. For example, the trigonometric equation. So, the answer to the equation 2sin2(2x)+5sin(2x)-3=0 is x=π/4+πk, where k is an integer. That is, if you substitute any integer values of the parameter k the argument x will satisfy the given equation.

5

In trigonometric problems, you may need to find all the negative roots, or the maximum of the negative. In solving such problems, apply logical reasoning or the method of mathematical induction. Several substitute integer values for k into the expression x=π/4+πk and see how it behaves argument. By the way, the largest negative root in the previous equation will be x=-3π/4 for k=1.