Whakaoti i te {l}{text{Losop.}}{text{a}(2x-5)^2-(x+1)^2=7x}

$$\left. \begin{array} { l } { \text { Los op. } } \\ { \text { a } ( 2 x - 5 ) ^ { 2 } - ( x + 1 ) ^ { 2 } = 7 x } \end{array} \right.$$

Whakaoti mō x

$$x = \frac{\sqrt{553} + 29}{6} \approx 8.752658672$$

$$x=\frac{29-\sqrt{553}}{6}\approx 0.914007995$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2x-5\right)^{2}.

$$4x^{2}-20x+25-\left(x+1\right)^{2}=7x$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$4x^{2}-20x+25-\left(x^{2}+2x+1\right)=7x$$

Hei kimi i te tauaro o x^{2}+2x+1, kimihia te tauaro o ia taurangi.

$$4x^{2}-20x+25-x^{2}-2x-1=7x$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}-20x+25-2x-1=7x$$

Pahekotia te -20x me -2x, ka -22x.

$$3x^{2}-22x+25-1=7x$$

Tangohia te 1 i te 25, ka 24.

$$3x^{2}-22x+24=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$3x^{2}-22x+24-7x=0$$

Pahekotia te -22x me -7x, ka -29x.

$$3x^{2}-29x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -29 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 3\times 24}}{2\times 3}$$

Pūrua -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\times 3\times 24}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-29\right)±\sqrt{841-12\times 24}}{2\times 3}$$

Whakareatia -12 ki te 24.

$$x=\frac{-\left(-29\right)±\sqrt{841-288}}{2\times 3}$$

Tāpiri 841 ki te -288.

$$x=\frac{-\left(-29\right)±\sqrt{553}}{2\times 3}$$

Ko te tauaro o -29 ko 29.

$$x=\frac{29±\sqrt{553}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{29±\sqrt{553}}{6}$$

Nā, me whakaoti te whārite x=\frac{29±\sqrt{553}}{6} ina he tāpiri te ±. Tāpiri 29 ki te \sqrt{553}.

$$x=\frac{\sqrt{553}+29}{6}$$

Nā, me whakaoti te whārite x=\frac{29±\sqrt{553}}{6} ina he tango te ±. Tango \sqrt{553} mai i 29.

$$x=\frac{29-\sqrt{553}}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{553}+29}{6} x=\frac{29-\sqrt{553}}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2x-5\right)^{2}.

$$4x^{2}-20x+25-\left(x+1\right)^{2}=7x$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$4x^{2}-20x+25-\left(x^{2}+2x+1\right)=7x$$

Hei kimi i te tauaro o x^{2}+2x+1, kimihia te tauaro o ia taurangi.

$$4x^{2}-20x+25-x^{2}-2x-1=7x$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}-20x+25-2x-1=7x$$

Pahekotia te -20x me -2x, ka -22x.

$$3x^{2}-22x+25-1=7x$$

Tangohia te 1 i te 25, ka 24.

$$3x^{2}-22x+24=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$3x^{2}-22x+24-7x=0$$

Pahekotia te -22x me -7x, ka -29x.

$$3x^{2}-29x+24=0$$

Tangohia te 24 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x^{2}-29x=-24$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-29x}{3}=-\frac{24}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{29}{3}x=-\frac{24}{3}$$

Whakawehe -24 ki te 3.

$$x^{2}-\frac{29}{3}x=-8$$

Whakawehea te -\frac{29}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{29}{6}. Nā, tāpiria te pūrua o te -\frac{29}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{29}{3}x+\left(-\frac{29}{6}\right)^{2}=-8+\left(-\frac{29}{6}\right)^{2}$$

Pūruatia -\frac{29}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{29}{3}x+\frac{841}{36}=-8+\frac{841}{36}$$

Tāpiri -8 ki te \frac{841}{36}.

$$x^{2}-\frac{29}{3}x+\frac{841}{36}=\frac{553}{36}$$

Tauwehea x^{2}-\frac{29}{3}x+\frac{841}{36}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{6}\right)^{2}=\frac{553}{36}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{29}{6}\right)^{2}}=\sqrt{\frac{553}{36}}$$

Whakarūnātia.

$$x-\frac{29}{6}=\frac{\sqrt{553}}{6} x-\frac{29}{6}=-\frac{\sqrt{553}}{6}$$

Me tāpiri \frac{29}{6} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{553}+29}{6} x=\frac{29-\sqrt{553}}{6}$$