f(x)=-2x^2+8x+4-এর সমাধান করুন

$$f ( x ) = - 2 x ^ { 2 } + 8 x + 4$$

Factor

$$-2\left(x-\left(2-\sqrt{6}\right)\right)\left(x-\left(\sqrt{6}+2\right)\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-2x^{2}+8x+4=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-2\right)\times 4}}{2\left(-2\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-2\right)\times 4}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-8±\sqrt{64+8\times 4}}{2\left(-2\right)}$$

Multiply 8 times 4.

$$x=\frac{-8±\sqrt{64+32}}{2\left(-2\right)}$$

Add 64 to 32.

$$x=\frac{-8±\sqrt{96}}{2\left(-2\right)}$$

Take the square root of 96.

$$x=\frac{-8±4\sqrt{6}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-8±4\sqrt{6}}{-4}$$

Now solve the equation x=\frac{-8±4\sqrt{6}}{-4} when ± is plus. Add -8 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}-8}{-4}$$

Divide -8+4\sqrt{6} by -4.

$$x=2-\sqrt{6}$$

Now solve the equation x=\frac{-8±4\sqrt{6}}{-4} when ± is minus. Subtract 4\sqrt{6} from -8.

$$x=\frac{-4\sqrt{6}-8}{-4}$$

Divide -8-4\sqrt{6} by -4.

$$x=\sqrt{6}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2-\sqrt{6} for x_{1} and 2+\sqrt{6} for x_{2}.

$$-2x^{2}+8x+4=-2\left(x-\left(2-\sqrt{6}\right)\right)\left(x-\left(\sqrt{6}+2\right)\right)$$

Evaluate

$$4+8x-2x^{2}$$