# Math Problem Solver With Steps

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## Solved Examples

Question 1: Differentiate y = (2x2 + 6)3
Solution:
Given y = (2x2 + 6)3

Differentiate y with respect to x

$\frac{dy}{dx}$ = $\frac{d}{dx}$ (2x2 + 6)3

$\frac{dy}{dx}$ = 3(2x2 + 6) $\frac{d}{dx}$(2x2 + 6)

(by using chain rule)

= 3(2x2 + 6)(4x)

= 12x(2x2 + 6)

= 24x3 + 72x

=> $\frac{dy}{dx}$ = 24x3 + 72x

Question 2: Solve $\sqrt{4 - 7x} = \sqrt{2}x$

Solution:
Given $\sqrt{4 - 7x} = \sqrt{2}x$

Step 1:

Squaring both side to remove the radical

=> $(\sqrt{4 - 7x})^2 = ( \sqrt{2}x)^2$

=> 4 - 7x = 2x2

=> 2x2 + 7x - 4 = 0

which is quadratic equation

Step 2:

Solve the quadratic equation

2x2 + 7x - 4 = 0

=> 2x2 + 8x - x - 4 = 0

=> 2x(x + 4) - (x + 4) = 0

=> (2x - 1)(x + 4) = 0

Step 3:

either 2x - 1 = 0       or      x + 4 = 0

if 2x - 1 = 0

=> 2x = 1

=> x = $\frac{1}{2}$

if x + 4 = 0

=> x = - 4

Answer: x = - 4, $\frac{1}{2}$

Question 3: Find all real solution to the rational equation.
$\frac{2}{x - 1} - \frac{1}{x + 2}$ = 1

Solution:
Given, rational equation is $\frac{2}{x - 1} - \frac{1}{x + 2}$ = 1

Step 1:

The LCM of the denominators of the rational expressions is

LCM = (x - 1)(x + 2)

Step 2:
Now multiply both sides of the equations by the LCM and simplify

=> $\frac{2 (x - 1)(x + 2)}{x - 1} - \frac{1 (x - 1)(x + 2)}{x + 2}$ = 1 * (x - 1)(x + 2)

=> 2(x + 2) - (x - 1) = x(x + 2) - 1(x + 2)

=> 2x + 4 - x + 1 = x2 + 2x - x - 2

=>  x + 5 = x2 + x - 2

=> x2 + x - 2 - x - 5 = 0

=> x2 - 7 = 0

=> x2 = 7

=> x = $\pm\sqrt{7}$

The solutions are, x = $- \sqrt{7}$  and  $\sqrt{7}$.