Whakaoti i te (x-3/2)^2+2x(x-1/2)<3(x^2+1/4)

$$( x - \frac { 3 } { 2 } ) ^ { 2 } + 2 x ( x - \frac { 1 } { 2 } ) < 3 ( x ^ { 2 } + \frac { 1 } { 4 } )$$

Whakaoti mō x

$$x>\frac{3}{8}$$

Ngā Upane Otinga

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-\frac{3}{2}\right)^{2}.

$$x^{2}-3x+\frac{9}{4}+2x\left(x-\frac{1}{2}\right)<3\left(x^{2}+\frac{1}{4}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x-\frac{1}{2}.

$$x^{2}-3x+\frac{9}{4}+2x^{2}-x<3\left(x^{2}+\frac{1}{4}\right)$$

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

$$3x^{2}-3x+\frac{9}{4}-x<3\left(x^{2}+\frac{1}{4}\right)$$

Pahekotia te -3x me -x, ka -4x.

$$3x^{2}-4x+\frac{9}{4}<3\left(x^{2}+\frac{1}{4}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x^{2}+\frac{1}{4}.

$$3x^{2}-4x+\frac{9}{4}<3x^{2}+\frac{3}{4}$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$3x^{2}-4x+\frac{9}{4}-3x^{2}<\frac{3}{4}$$

Pahekotia te 3x^{2} me -3x^{2}, ka 0.

$$-4x+\frac{9}{4}<\frac{3}{4}$$

Tangohia te \frac{9}{4} mai i ngā taha e rua.

$$-4x<\frac{3}{4}-\frac{9}{4}$$

Tangohia te \frac{9}{4} i te \frac{3}{4}, ka -\frac{3}{2}.

$$-4x<-\frac{3}{2}$$

Whakawehea ngā taha e rua ki te -4. I te mea he tōraro a -4, ka huri te ahunga koreōrite.

$$x>\frac{-\frac{3}{2}}{-4}$$

Tuhia te \frac{-\frac{3}{2}}{-4} hei hautanga kotahi.

$$x>\frac{-3}{2\left(-4\right)}$$

Whakareatia te 2 ki te -4, ka -8.

$$x>\frac{-3}{-8}$$

Ka taea te hautanga \frac{-3}{-8} te whakamāmā ki te \frac{3}{8} mā te tango tahi i te tohu tōraro i te taurunga me te tauraro.

$$x>\frac{3}{8}$$