How to solve a quadratic inequality in detail

Let's be clear. 2x+1=9.

2x=8x=4...

An inequality with an unknown number and the highest order of the unknown number is 2th degree is called a unary quadratic inequality, and its general form is ax 2+bx+c>0 or ax 2+bx+c<0 (a is not equal to 0), where ax 2+bx+c quadratic trinomials on the field of real numbers.

Solution of unary quadratic inequality 1) When v("v"Indicates that when the discriminant is, the same below) = b 2-4ac> = 0, the quadratic trinomial, ax 2+bx+c has two real roots, then ax 2+bx+c can always be decomposed into the form a(x-x1)(x-x2). In this way, solving a quadratic inequality boils down to solving a group of two unary quadratic inequalities. The solution set of a unary quadratic inequality is the union of the solution set of these two groups of unary inequalities.

Let's take an example.

2x^2-7x+6<0

Make use of cross multiplication.

(2x-3) (x-2) <0

Then, it is discussed in two cases:

1. 2x-3<0, x-2>0

Get x" and x>2. Not true.

2. 2x-3>0, x-2<0

Get x" and x>2.

The solution set of the final inequality is:

x<2 and x>

The set of solutions to the inequality is .

Convert it to y x 2-3x 1

Then make an image of it.

Observing the image when the value of y is greater than 0 is the value of x, which is the second way to solve the quadratic inequality:

Let the right side of the quadratic inequality be 0

For example: x 2-3x 1>0

Reams x 2-3x 1 0

This quadratic equation is then factored.

Converted to (a+b)*(c+d)=0.

This is brought into the original inequality.

For example, when (a+b)*(c+d)>0.

Both items are greater than 0 or both are less than 0 to solve.

The answer can also be solved.

The picture is not easy to draw, so I can only explain it like this, I don't know if I can understand it.

Unary quadratic inequalityThere are several solutions:

1. When -=b3-4ac 0, quadratic trinomial, ax2+bx+c has two real roots, then ax2+bx+c, can always be decomposed into the form a(x-x1)(x-x2). In this way, the solution-element quadratic inequality can be reduced to solving two groups of unary quadratic inequalities. The solution set of a one-dimensional quadratic inequality is the intersection of the solution set of these two one-dimensional inequalities.

2. Use the matching method.

Solution-element quadratic inequality.

3. Image through a one-dimensional quadratic function.

Solve the two intersections of the Xintan, the image of the second slippery function and the x-axis, and then according to the requirements of the problem"<0"or">0"And the answer is introduced.

4. a number axis. Penetrating the root: When solving the higher inequality using the root axis method, the inequality - end is first reduced to zero, and then the other end is factored.

And find the nucleus rock out of its zero points, mark these zero points on the number axis, and then use a smooth curve, from the right end of the x-axis, through these zero points in turn.

The solution of the inequality greater than zero corresponds to the set of values of the real number x of the part of the curve above the x-axis, and the opposite is true for those less than zero. This method is called the sequential shaft root method.

How to solve a system of quadratic equations.

First of all, when a is not equal to 0, the equation: ax 2+bx+c=0 is a quadratic equation.

1.Formula method: Tanyou δ=b -4ac, the equation is unsolved when δ 0, and δ 0.

x=[-b under the root number (b -4ac)] 2a (there is only one x when δ=0).

2.Matching method: The equation can be reduced to [x-(-b 2a)] b -4ac) 4a

It can be solved: x=[-b root number (b -4ac)] 2a (from which the formula method is derived).

3.The direct leveling method is similar to the matching method.

4.Factorization method: Nuclear fission Qingxin is, of course, factorization, take a look at this equation.

ax+c)(bx+d)=0, ABX +(AD+BC)+CD=0 is compared with the unary quadratic equation ax 2+bx+c=0 to obtain a=ab, b=ad+bc, c=cd. The so-called factorization is nothing more than finding the four numbers a, b, c, and d.

Let's give you a few examples.

Example 1: x -5x+6=0

Solution: (x-2)(x-3)=0,x1=2,x2=3

Example 2: 3x -17x+10=0

Solution: (3x-2)(x-5)=0,x1=2 3,x2=5

The factoring method, also known as the cross multiplication method, is known for the reasons below.

abx²+（ad+bc）+cd=0 ax c

bx d (a, b, c, d are not necessarily all positive).

When solving equations, choose the appropriate method.

Here are a few practice questions to try.

How to solve the unary quadratic inequality of two unknowns.

For example: a -4>0

a^>4

a> plus or minus 2

Commentary: When solving a quadratic inequality, such as an appeal problem, move the term without an unknown number first, and when eliminating an equation, you must do the opposite operation sign to it, such as adding it when it is subtraction; When it is a division, divide and so on. In the example, if it is squared, it should be squared.

When 4 is squared, it should be noted that it is plus or minus 2. Note: When dividing by a negative number, change the sign.

The rest is to do more examples of one-dimensional quadratic inequalities, and if you do more, you will naturally master some methods, and if you have any doubts, you can also ask others until you understand them.

Steps to solve a quadratic inequality of one element:

Taking the number line root penetration method as an example, the steps to solve the unary quadratic inequality are as follows: 1. Turn the quadratic coefficient into positive; 2. Draw the number line, and mark all the roots in order from small to large on the number line; 3. Starting from the upper right corner, go through the root of the inequality one by one, and cross the odd power when encountering the term containing x, and cross it to the even power; 4. Pay attention to the root that makes the inequality 0.

Definition of unary quadratic inequalities.

Unary quadratic inequality refers to a non-bichai equation that contains an unknown number and the highest order of the unknown number is 2, which is called a unary quadratic inequality. Its general form is ax +bx+c>0, ax +bx+c≠0, ax +bx+c<0 (a does not equal 0).

Further reading: Discriminant methods for quadratic inequalities.

1) When a>0.

When discriminant =b -4ac>0, ax +bx+c=0 are two unequal real roots (let the solution of x10 be xx2.

When the discriminant formula =b -4ac=0, because a>0, the opening of the quadratic function image is upward, and the parabola has an intersection point with the x-axis, x1=x2, so the solution of the inequality ax +bx+c>0 is the whole real number of x≠x1, and the solution set of the inequality ax +bx+c<0 is an empty set.

When the discriminant formula =b -4ac<0, the parabola has no intersection point with the x-axis above the x-axis, so the solution set of the inequality ax +bx+c>0 is the pre-elimination whole real number, and the solution set of the inequality ax +bx+c<0 is an empty set, that is, there is no solution.

2) When a<0.

When discriminant =b -4ac>0, ax +bx+c=0 are two unequal real roots (let the solution of x10 be x1

When the discriminant formula =b -4ac=0, because a<0, the opening of the quadratic function image is downward, the parabola has an intersection point with the x-axis, x1=x2, so the solution of the inequality ax +bx+c<0 is the whole real number of x≠x1, and the solution set of the inequality ax +bx+c>0 is an empty set.

When the discriminant = b -4ac<0, the parabola has no intersection with the x-axis above the x-axis, so the solution set of the inequality ax +bx+c<0 is the whole real number, and the solution set of the inequality ax +bx+c>0 is an empty set, i.e., there is no solution.

The unary quadratic inequalities ax 2 ten bx ten c>0 and ax 2 + bx ten c<0 when the equation ax 2 ten bx ten c = 0 has a solution to the inequality ax 2 ten bx + c<0 when 0 is the whole real number. When b 2 a 4ac > 0 ax 2 ten bx ten c > 0 the solution is a>0 x is larger than the larger equation root or less than the smaller equation root, and a "destroy the head with 0 x is greater than the smaller equation root and less than the larger equation root.

When the number axis penetrating method uses the root axis method to solve the higher order inequality, it is to first reduce one end of the inequality to zero, then decompose the factor to the other end, and find its zero points, mark these zero points on the number axis, and then use a smooth curve, starting from the right end of the x axis, through these zero points in turn, the solution of the inequality greater than zero corresponds to the real number x value set of the trembling part of the curve above the x axis, and the opposite is less than zero.

For example, the inequality: x -3x+2 0 (the highest minor coefficient must be positive, not positive to be positive).

Decomposition factor: (x-1)(x-2) 0;

Find the root ridge book of equation (x-1) (x-2) = 0: x=1 or x=2;

Draw the number axis and mark the point where the root Sakura Cave is macro;

Note that at this time, starting from the far right, a curve is drawn from the upper right of point 2, passing through point 2, continuing to draw to the left, similar to a parabola, and then passing through point 1, extending infinitely to the upper left of point 1;

Look at the problem to solve, the problem requires the solution of 0, then you only need to see which section is on the number line and below the number line, and you can observe: 1 x 2.

Definition of knowledge points**&&ExplanationA quadratic inequality is an inequality of a polynomial of order 2, usually written in the form ax 2+bx+c>0 (or <0). Refers to the discriminant expression of a quadratic equation, i.e., =b 2-4ac.

Application of knowledge pointsWhen solving a quadratic inequality, it is necessary to use the value of to find the root of the equation, and then judge the solution set of the inequality according to the range of the root. When >0, there are two unequal real roots, and the solution set of the inequality is x< or x> ; When =0, there are two equal real roots, and the solution set of the inequality is x= ; When <0, there is no real root, that is, there is no solution to the inequality.

Explanation of knowledge points and example questions:

For example, given an inequality x 2+2x-3 > 0, you want to judge its solution set. First, find =2 2-4 1 (-3)=16, and since >0, the equation has two unequal real roots. Seek out the root to get:

x1=(-2+ 16) 2=1, x2=(-2- 16) 2=-3, so the solution set of the inequality is x<-3 or x>1.

Move all the formulas aside. Solve the equation on one side. Two roots are derived.

Then mark the coordinate system with two roots. Make an image of the equation. See what the area is greater than 0 and what the area is less than 0.

Then look at whether the formula is greater than 0 or less than 0

Think of x 2+2x as a quadratic function.

Let y=x 2+2x

Thus, the solutions of the inequality x 2+2x 0 are those x's of y>0, those abscissa x's of the ordinate y>0, and the abscissa x corresponding to y=x 2+2x parabola above the x-axis (ordinate y>0).

The parabola y=x2+2x intersects the x-axis at two points (-2,0) and (0,0), so x<-2 or x>0

The problem is very simple, but this kind of problem-solving idea is very classic! It should not be limited to this question, but should experience the idea of combining numbers and shapes! The solution is exactly what everyone said.