Question

plz explain me this
I am totally confused ​

Answer 1

Question :-

If $\bf{x = \dfrac{\sqrt{3} + 1}{2}}$ , find the value of $\bf{4x^{3} + 2x^{2} - 8x + 7}$

Solution :-

By substituting the value of x in the equation , we get :-

$:\implies \bf{4x^{3} + 2x^{2} - 8x + 7}$

$:\implies \bf{4 \times \bigg(\dfrac{\sqrt{3} + 1}{2}\bigg)^{3} + 2 \times \bigg(\dfrac{\sqrt{3} + 1}{2}\bigg)^{2} - 8 \times \bigg(\dfrac{\sqrt{3} + 1}{2}\bigg) + 7}$

$\boxed{\begin{minipage}{7 cm}Using the identities and substituting them in the equation, we get :-\\ \\ \underline{\bf{(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)}} \\ \\ \underline{\bf{(a + b)^{2} = a^{2} + b^{2} + 2ab}} \end{minipage}}$

$:\implies \bf{4 \times \bigg(\dfrac{(\sqrt{3})^{3} + 1^{3} + 3 \times \sqrt{3} \times 1(\sqrt{3} + 1)}{2^{3}}\bigg)}$$\bf{2 \times \bigg(\dfrac{(\sqrt{3})^{2} + 1^{2} + 2 \times \sqrt{3} \times 1}{2^{2}}\bigg) - 8 \times \bigg(\dfrac{\sqrt{3} + 1}{2}\bigg) + 7}$

$:\implies \bf{4 \times \bigg(\dfrac{3\sqrt{3} + 1 + 3\sqrt{3}(\sqrt{3} + 1)}{8}\bigg) + 2 \times \bigg(\dfrac{3 + 1 + 2\sqrt{3}}{4}\bigg) - 8 \times \bigg(\dfrac{\sqrt{3} + 1}{2}\bigg) + 7}$

$:\implies \bf{\not{4} \times \bigg(\dfrac{3\sqrt{3} + 1 + 3\sqrt{3}(\sqrt{3} + 1)}{\not{8}}\bigg) + \not{2} \times \bigg(\dfrac{3 + 1 + 2\sqrt{3}}{4}\bigg) - \not{8} \times \bigg(\dfrac{\sqrt{3} + 1}{\not{2}}\bigg) + 7}$

$:\implies \bf{\bigg(\dfrac{3\sqrt{3} + 1 + 3\sqrt{3}(\sqrt{3} + 1)}{2}\bigg) + \bigg(\dfrac{3 + 1 + 2\sqrt{3}}{2}\bigg) - 4 \times \sqrt{3} + 1 + 7}$

$:\implies \bf{\bigg(\dfrac{3\sqrt{3} + 1 + 9 + 3\sqrt{3}}{2}\bigg) + \bigg(\dfrac{4 + 2\sqrt{3}}{2}\bigg) - 4 \times \sqrt{3} + 8}$

$:\implies \bf{\bigg(\dfrac{3\sqrt{3} + 1 + 9 + 3\sqrt{3}}{2}\bigg) + \bigg(\dfrac{4 + 2\sqrt{3}}{2}\bigg) - 4 \times (\sqrt{3} + 1) + 7}$

$:\implies \bf{\bigg(\dfrac{6\sqrt{3} + 10}{2}\bigg) + \bigg(\dfrac{4 + 2\sqrt{3}}{2}\bigg) - 4\sqrt{3} - 4 + 7}$

$:\implies \bf{\bigg(\dfrac{6\sqrt{3} + 10 + 4 + 2\sqrt{3}}{2}\bigg) - 4\sqrt{3} - 3}$

$:\implies \bf{\bigg(\dfrac{8\sqrt{3} + 14 - 8\sqrt{3} - 6}{2}\bigg)}$

$:\implies \bf{\bigg(\dfrac{14 - 6}{2}\bigg)}$

$:\implies \bf{\bigg(\dfrac{8}{2}\bigg)}$

$:\implies \bf{\dfrac{\not{8}}{\not{2}}}$

$:\implies \bf{4}$

$\therefore \purple{\bf{4x^{3} + 2x^{2} - 8x + 7 = 4}}$

Hence, the value of 4x³ + 2x² - 8x + 7 is 4.