The unraveling of irregular square plains. Connection between modules
In this article we will look at the unraveling of irregular square levels.
Let’s repeat from the very beginning, as they are called square. Similar to the form ax 2 + bx + c = 0, de x is changeable, and the coefficients a, b and tens of numbers, and a ≠ 0 is called square. Just as the coefficient at x 2 is not equal to zero, the coefficient at x and another term can be equal to zero, in which case they are not necessarily equal to the square.
Uneven square plains come in three types:
1) If b = 0, s ≠ 0, then ax 2 + c = 0;
2) If b ≠ 0, c = 0, then ax 2 + bx = 0;
3) If b = 0, c = 0, then ax2 = 0.
- Let's figure out how to be respectful respect for the form ah 2+c=0.
To untie the relationship, we move the right member from the right side of the relationship, we remove it
ax 2 = ‒s. If a ≠ 0, then we can separate the parts equal to a, then x 2 = ‒c/a.
If ‒с/а > 0, then the equation has two roots
x = ±√(-c/a) .
Yaksho w ‒c/a< 0, то это уравнение решений не имеет. Более наглядно решение данных уравнений представлено на схеме.
Let’s try to get used to the butts, as there is such jealousy.
Butt 1. Untie the row 2x 2 ‒ 32 = 0.
Example: x 1 = ‒ 4, x 2 = 4.
Butt 2. Unravel the equation 2x 2 + 8 = 0.
The answer: there is no solution to jealousy.
- Let's figure out how to lie equal to the form ax 2+bx = 0.
To unravel the equation ax 2 + bx = 0, we decompose it into multipliers, so that it is carried by the arms of x, and we subtract x (ax + b) = 0. The addition is equal to zero, since we want one of the multipliers to be equal to zero. Then either x = 0, or ax + b = 0. The highest value is ax + b = 0, we reject ax = b, the star x = b/a. Following the form ax 2 + bx = 0, there are two roots x 1 = 0 and x 2 = b/a. Marvel at how the solution looks like this on the diagram. 
Let's consolidate our knowledge on a specific application.
Butt 3. Unravel the equation 3x 2 – 12x = 0.
x(3x ‒ 12) = 0
x = 0 or 3x - 12 = 0
Example: x1 = 0, x2 = 4.
- Rivnyannya of the third type ah 2 = 0 It’s very easy to be admired.
If x 2 = 0, then x 2 = 0. There are two equal roots x 1 = 0, x 2 = 0.
For clarity, let's look at the diagram. 
We turn over to the top of the butt 4, so that equalization of this type is even simpler.
butt 4. Unravel the level 7x2 = 0.
Example: x 1, 2 = 0.
It didn’t immediately dawn on us what kind of uneven square alignment we should face. Let's take a look at the offensive butt.
Butt 5. Virishity Rivalry
Let's multiply the offending parts of the jealousy by the Zagalny banner, then by 30
Let's quickly
5 (5x2 + 9) - 6 (4x 2 - 9) = 90.
Opening the temples
25x2 + 45 - 24x 2 + 54 = 90.
Let's find out more
Moved 99 from the left side to the right side, changing the sign to the opposite one
Proof: the root is silent.
We learned how uneven square planes work. I hope that now you will not have problems with such tasks. Be respectful of the appearance of the irregular square surface, and then everything will work out for you.
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We present to you a truly catless online calculator for solving square levels. You can quickly remove and smell the stench that lingers on large butts.
Make some money solution to square renovation online, from now on, bring the jealousy to an obscure look:
ax 2 + bx + c = 0
Fill in each form field:
How to balance squarely
| How to square the square: | Types of roots: |
|
1.
Bring the square down to a glamorous look: Zagalny Viglyad Аx2+Bx+C=0 Stock: 3x - 2x2+1=-1 Aimable to -2x2+3x+2=0 2.
Known discriminant D. 3.
We know the root of the verse. |
1.
The correct root. Moreover. x1 is not the same as x2 The situation is worse if D>0 and A are not equal to 0. 2.
The correct root is avoided. x1 and x2 3.
Two complex roots. x1=d+ei, x2=d-ei, de i=-(1) 1/2 5.
Rivalry is a faceless decision. 6.
There is no decision to be made. |
To consolidate the algorithm, ostentatious butts of ties of square levels.
Example 1. Version of the original square equation with different active roots.
x 2 + 3x -10 = 0
Whose equals
A = 1, B = 3, C = -10
D=B 2 -4*A*C = 9-4*1*(-10) = 9+40 = 49
The square root will be designated as the number 1/2!
x1=(-B+D 1/2)/2A = (-3+7)/2 = 2
x2=(-B-D 1/2)/2A = (-3-7)/2 = -5
For verification we substitute:
(x-2)*(x+5) = x2 -2x +5x - 10 = x2 + 3x -10
Example 2. Unraveling a square row from the escape of active roots.
x 2 - 8x + 16 = 0
A = 1, B = -8, C = 16
D = k 2 - AC = 16 - 16 = 0
X = -k/A = 4
Let's imagine
(x-4) * (x-4) = (x-4) 2 = X 2 - 8x + 16
Example 3. Connecting a square line with complex roots.
13x 2 - 4x + 1 = 0
A=1, B=-4, C=9
D = b 2 - 4AC = 16 - 4 * 13 * 1 = 16 - 52 = -36
The negative discriminant is a more complex root.
X1=(-B+D 1/2)/2A = (4+6i)/(2*13) = 2/13+3i/13
x2=(-B-D 1/2)/2A = (4-6i)/(2*13) = 2/13-3i/13
de I – ce square root z -1
Axis, power, all possible consequences of unleashing the square levels.
We hope that our online calculator appear even brighter for you.
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Square level.
Square Rivnyanya- algebraic equal to the literal view
de x - free change,
a, b, c, - coefficients, and
Viraz 
called a quadratic trinomial.
Methods for untying square rows.
1. WAY : Decompose the left side of the row into multipliers.
Let's unleash jealousy x 2 + 10x - 24 = 0. Let's split the left part into multipliers:
x 2 + 10x - 24 = x 2 + 12x - 2x - 24 = x (x + 12) - 2 (x + 12) = (x + 12) (x - 2).
Well, jealousy can be rewritten like this:
(x + 12) (x - 2) = 0
Since the addition is equal to zero, then we take one of the multipliers equal to zero. Therefore, the left side of the equation goes to zero when x = 2, and also with x = - 12. Tse means the number 2 і - 12 e respect for the roots x 2 + 10x - 24 = 0.
2. WAY : The method of seeing a full square.
Let's unleash jealousy x 2 + 6x - 7 = 0. Apparently the left side has a new square.
For this we write viraz x 2 + 6x in the next view:
x 2 + 6x = x 2 + 2 x 3.
In the extracted form, the first addition is the square of the number x, and the other is the double addition of x by 3. Therefore, to remove the second square, you need to add 3 2 so
x 2 + 2 x 3 + 32 = (x + 3) 2.
Let us now reconstitute the left part of the equation
x 2 + 6x - 7 = 0,
adding to it and going beyond it 3 2 . Maemo:
x 2 + 6x - 7 = x 2 + 2 x 3 + 3 2 - 3 2 - 7 = (x + 3) 2 - 9 - 7 = (x + 3) 2 - 16.
In this manner, this reverence can be written like this:
(x + 3) 2 – 16 = 0, (x + 3) 2 = 16.
Otje, x + 3 - 4 = 0, x 1 = 1, or x + 3 = -4, x 2 = -7.
3. WAY :The connection of square lines to the formula.
Multiplying the offending parts of the relationship
ax 2 + bx + c = 0, a ≠ 0
on 4a and we do it sequentially:
4a 2 x 2 + 4abx + 4ac = 0,
((2ax) 2 + 2ax b + b 2) - b 2 + 4ac = 0,
(2ax + b) 2 = b 2 - 4ac,
2ax + b = ± √ b 2 - 4ac,
2ax = - b ± √ b 2 - 4ac,
Apply it.
A) Let's talk about jealousy: 4x2+7x+3=0.
a = 4, b = 7, c = 3, D = b 2 - 4ac = 7 2 - 4 4 3 = 49 - 48 = 1,
D > 0, two different roots;
Well, once there is a positive discriminant, then. at
b 2 - 4ac >0, Rivnyanya ax 2 + bx + c = 0 There are two different roots.
b) Let's talk about jealousy: 4x 2 - 4x + 1 = 0,
a = 4, b = - 4, c = 1, D = b 2 - 4ac = (-4) 2 - 4 4 1 = 16 - 16 = 0,
D = 0, one root;
Well, if the discriminant is equal to zero, then. b 2 - 4ac = 0, then jealousy
ax 2 + bx + c = 0 there is a single root,
V) Let's talk about jealousy: 2x 2 + 3x + 4 = 0,
a = 2, b = 3, c = 4, D = b 2 - 4ac = 3 2 - 4 2 4 = 9 - 32 = - 13, D< 0.
There is no root truth.
Well, if the discriminant is negative, then. b 2 - 4ac< 0 , Rivnyanya
ax 2 + bx + c = 0 the root doesn't matter.
Formula (1) square roots ax 2 + bx + c = 0 allows you to know the root come what may square level (as there is a stench), including the induced and uneven one. Verbally, formula (1) looks like this: The square root is equal to the fraction, the number of which is equal to the other coefficient taken with the last sign, plus minus the square root of the square of which coefficient without the quadruple addition of the first coefficient to the third member, and the flag bearer is the sub-military first coefficient.
4. METHOD: The connection is made with the results of Viet's theorem.
As you can see, the square alignment appears
x 2 + px + c = 0.(1)
I am fundamentally satisfied with Viet’s theorem, as when a = 1 I can see
x 1 x 2 = q,
x 1 + x 2 = - p
You can create the following symbols (behind the coefficients p and q you can transfer the signs of the roots).
a) What is the member of the institution? q induced jealousy (1) positive ( q > 0), then the verse is two however behind the sign of the root and at the same time behind the other coefficient p. Yakshcho R< 0 , then resentment has negative roots, as R< 0 , then resentment's roots are positive.
For example,
x 2 - 3x + 2 = 0; x 1 = 2і x 2 = 1, so yeah q = 2 > 0і p = - 3< 0;
x 2 + 8x + 7 = 0; x 1 = - 7і x 2 = - 1, so yeah q = 7 > 0і p=8>0.
b) I am a free member q induced jealousy (1) negative ( q< 0 ), then there are two differences behind the sign of the root, and the larger root behind the modulus will be positive, since p< 0 , or negative, either p > 0 .
For example,
x 2 + 4x - 5 = 0; x 1 = - 5і x 2 = 1, so yeah q= - 5< 0 і p = 4> 0;
x 2 - 8x - 9 = 0; x 1 = 9і x 2 = - 1, so yeah q = - 9< 0 і p = - 8< 0.
apply it.
1) Rivalry is unconnected 345x 2 - 137x - 208 = 0.
Decision. So yak a + b + c = 0 (345 - 137 - 208 = 0), That
x 1 = 1, x 2 = c/a = -208/345.
Type: 1; -208/345.
2) Virishimo jealousy 132x 2 - 247x + 115 = 0.
Decision. So yak a + b + c = 0 (132 - 247 + 115 = 0), That
x 1 = 1, x 2 = c/a = 115/132.
Type: 1; 115/132.
B. Like another co-respondent b = 2k– the guy is a number, then the formula is root
butt.
Let's unleash jealousy 3x2 - 14x + 16 = 0.
Decision. Maemo: a = 3, b = - 14, c = 16, k = - 7;
D = k 2 - ac = (- 7) 2 - 3 16 = 49 - 48 = 1, D > 0, two different roots;
Type: 2; 8/3
Art. Rivalry has been established
x 2 + px + q = 0
avoids jealousy in a zagal manner, in which a = 1, b = pі c = q. Therefore, for the induced square equation, the formula of the roots is
I see:
Formula (3) is especially handy if R- guy's number.
butt. Let's unleash jealousy x 2 - 14x - 15 = 0.
Decision. Maemo: x 1.2 = 7±
Subject: x 1 = 15; x 2 = -1.
5. METHOD: The connection between the levels is more graphic.
butt. Unravel the level x2 – 2x – 3 = 0.
Let us graph the function y = x2 - 2x - 3
1) Maєmo: a = 1, b = -2, x0 = = 1, y0 = f (1) = 12 - 2 - 3 = -4. This means that the vertex of the parabola is the point (1; -4), and the entire parabola is the straight line x = 1.
2) We take two points on the x axis, symmetrical to the parabola axis, for example, points x = -1 and x = 3.
Let's say f(-1) = f(3) = 0. Let's stay on the coordinate plane of the point (-1; 0) and (3; 0).
3) Through the points (-1; 0), (1; -4), (3; 0) a parabola is drawn (Fig. 68).
Roots x2 – 2x – 3 = 0 є abscise points across the parabola from all x; Well, the root equation is: x1 = - 1, x2 - 3.