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Raskite didžiausią sveikąjį lygties $$\sqrt {x^{2}-4\cdot x+12} = 3$$ sprendinį.

Sprendimas.

saknis( x^²- 4* x+12)  = 
3

saknis( x^²- 4* x+12) = 3$$\sqrt {x^{2}-4\cdot x+12}$$ = $$3$$

 (saknis( x^²- 4* x+12))^² =  3^²$$(\sqrt {x^{2}-4\cdot x+12})^{2}$$ = $$3^{2}$$

$${\normalsize (\sqrt {x^{2}-4\cdot x+12})^{2}}$$ = $${\normalsize (x^{2}-4\cdot x+12)}$$

 x^²- 4* x+12 =  3^²$$x^{2}-4\cdot x+12$$ = $$3^{2}$$

$${\normalsize 3^{2}}$$ = $${\normalsize 9}$$

 x^²- 4* x+12 = 9$$x^{2}-4\cdot x+12$$ = $$9$$

 x^²- 4* x+12-9 = 0$$x^{2}-4\cdot x+12-9$$ = $$0$$

$${\normalsize 12-9}$$ = $${\normalsize 3}$$

 x^²- 4* x+3 = 0$$x^{2}-4\cdot x+3$$ = $$0$$

$${\normalsize x^{2}-4\cdot x+3}$$ = $${\normalsize (x-3)\cdot (x-1)}$$

Paaiškinimas:

Kvadratinis trinaris $${\normalsize a\cdot x^{2}+b\cdot x+c}$$, kur
a = 1, b = -4, c = 3.
Diskriminantas $${\normalsize D = b^{2}-4\cdot a\cdot c = 16-12}$$ = 4.

x1 = $${\normalsize \frac{4+\sqrt {4}}{2\cdot 1} = \frac{4+2}{2} = \frac{2+1}{1} = \frac{3}{1}}$$ = 3
x2 = $${\normalsize \frac{4-\sqrt {4}}{2\cdot 1} = \frac{4-2}{2} = \frac{2-1}{1} = \frac{1}{1}}$$ = 1

 (x-3)* (x-1) = 0$$(x-3)\cdot (x-1)$$ = $$0$$

Didesnė šaknis x = 3.

Patikriname lygybę, kai x = 3:

saknis( x^²- 4* x+12)  = 
3

saknis( x^²- 4* x+12) = 3$$\sqrt {x^{2}-4\cdot x+12}$$ = $$3$$

Paaiškinimas:

Keitimas $${\normalsize x}$$ = $${\normalsize 3}$$.

saknis( 3^²- 4* 3+12) = 3$$\sqrt {3^{2}-4\cdot 3+12}$$ = $$3$$

$${\normalsize 3^{2}}$$ = $${\normalsize 9}$$

saknis(9- 4* 3+12) = 3$$\sqrt {9-4\cdot 3+12}$$ = $$3$$

$${\normalsize 4\cdot 3}$$ = $${\normalsize 12}$$

saknis(9-12+12) = 3$$\sqrt {9-12+12}$$ = $$3$$

$${\normalsize -12+12}$$ = $$0$$

saknis(9+0) = 3$$\sqrt {9+0}$$ = $$3$$

saknis(9) = 3$$\sqrt {9}$$ = $$3$$

$${\normalsize \sqrt {9}}$$ = $${\normalsize 3}$$

3 = 3$$3$$ = $$3$$

$$\sqrt {3^{2}-4\cdot 3+12}$$  = $$3$$

$$\sqrt {9-12+12}$$  = $$3$$

$$\sqrt {9}$$  = $$3$$

$$3$$  = $$3$$

saknis(3^2-4*3+12)  =  3

saknis(9-12+12)  =  3

saknis(9)  =  3

3  =  3

Lygybė teisinga, taigi, didžiausia sveika lygties šaknis ir bus x = 3.

Atsakymas: 3