Resolver 76=a(1.5-h)^3+k

$$76 = a ( 1.5 - h ) ^ { 3 } + k$$

Ebatzi: a (complex solution)

$$\left\{\begin{matrix}a=-\frac{k-76}{\left(1.5-h\right)^{3}}\text{, }&h\neq \frac{3}{2}\\a\in \mathrm{C}\text{, }&k=76\text{ and }h=\frac{3}{2}\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

\left(1.5-h\right)^{3} zabaltzeko, erabili Newton-en binomioa \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}.

$$76=a\left(3.375-6.75h+4.5h^{2}-h^{3}\right)+k$$

Erabili banaketa-propietatea a eta 3.375-6.75h+4.5h^{2}-h^{3} biderkatzeko.

$$76=3.375a-6.75ah+4.5ah^{2}-ah^{3}+k$$

Trukatu aldeak, aldagaiak ezkerraldean egon daitezen.

$$3.375a-6.75ah+4.5ah^{2}-ah^{3}+k=76$$

Kendu k bi aldeetatik.

$$3.375a-6.75ah+4.5ah^{2}-ah^{3}=76-k$$

Konbinatu a duten gai guztiak.

$$\left(3.375-6.75h+4.5h^{2}-h^{3}\right)a=76-k$$

Modu arruntean dago ekuazioa.

$$\left(-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375\right)a=76-k$$

Zatitu ekuazioaren bi aldeak 3.375-6.75h+4.5h^{2}-h^{3} balioarekin.

$$\frac{\left(-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375\right)a}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}=\frac{76-k}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}$$

3.375-6.75h+4.5h^{2}-h^{3} balioarekin zatituz gero, 3.375-6.75h+4.5h^{2}-h^{3} balioarekiko biderketa desegiten da.

$$a=\frac{76-k}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}$$

Zatitu 76-k balioa 3.375-6.75h+4.5h^{2}-h^{3} balioarekin.

$$a=\frac{8\left(76-k\right)}{\left(3-2h\right)^{3}}$$

Ebatzi: a

$$\left\{\begin{matrix}a=-\frac{k-76}{\left(1.5-h\right)^{3}}\text{, }&h\neq \frac{3}{2}\\a\in \mathrm{R}\text{, }&k=76\text{ and }h=\frac{3}{2}\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

\left(1.5-h\right)^{3} zabaltzeko, erabili Newton-en binomioa \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}.

$$76=a\left(3.375-6.75h+4.5h^{2}-h^{3}\right)+k$$

Erabili banaketa-propietatea a eta 3.375-6.75h+4.5h^{2}-h^{3} biderkatzeko.

$$76=3.375a-6.75ah+4.5ah^{2}-ah^{3}+k$$

Trukatu aldeak, aldagaiak ezkerraldean egon daitezen.

$$3.375a-6.75ah+4.5ah^{2}-ah^{3}+k=76$$

Kendu k bi aldeetatik.

$$3.375a-6.75ah+4.5ah^{2}-ah^{3}=76-k$$

Konbinatu a duten gai guztiak.

$$\left(3.375-6.75h+4.5h^{2}-h^{3}\right)a=76-k$$

Modu arruntean dago ekuazioa.

$$\left(-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375\right)a=76-k$$

Zatitu ekuazioaren bi aldeak 3.375-6.75h+4.5h^{2}-h^{3} balioarekin.

$$\frac{\left(-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375\right)a}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}=\frac{76-k}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}$$

3.375-6.75h+4.5h^{2}-h^{3} balioarekin zatituz gero, 3.375-6.75h+4.5h^{2}-h^{3} balioarekiko biderketa desegiten da.

$$a=\frac{76-k}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}$$

Zatitu 76-k balioa 3.375-6.75h+4.5h^{2}-h^{3} balioarekin.

$$a=\frac{8\left(76-k\right)}{\left(3-2h\right)^{3}}$$

Ebatzi: h (complex solution)

$$\left\{\begin{matrix}h=e^{\frac{i\times 4\pi }{3}}a^{-\frac{1}{3}}\sqrt[3]{k-76}+1.5\text{; }h=1.5+a^{-\frac{1}{3}}\sqrt[3]{k-76}\text{; }h=e^{\frac{i\times 2\pi }{3}}a^{-\frac{1}{3}}\sqrt[3]{k-76}+1.5\text{, }&a\neq 0\\h\in \mathrm{C}\text{, }&k=76\text{ and }a=0\end{matrix}\right.$$

Ebatzi: h

$$\left\{\begin{matrix}h=\frac{\sqrt[3]{k-76}+\frac{3\sqrt[3]{a}}{2}}{\sqrt[3]{a}}\text{, }&a\neq 0\\h\in \mathrm{R}\text{, }&k=76\text{ and }a=0\end{matrix}\right.$$