What substitution should be used to rewrite 6(x + 5)² + 5(x + 5) - 4 = 0 as a quadratic equation?

Solution:

Given: Equation is 6(x + 5)² + 5(x + 5) - 4 = 0

To solve the above equation by means of substitution we replace (x+5) by say, u

The equation becomes:

6u² + 5u - 4 = 0 (1)

Since this quadratic equation cannot be factorized, its roots can be found out by using the quadratic formula.

u = -b ± √b² - 4ac/2a

Hence the roots of equation (1) are

u= -5² ± √6² - 4(6)(-4)/2(6)

= -5 ± 2√33/12

= -5/12 ± (√33)/6

Since u = x + 5 = -5/12 ± (√33)/6

x = -65/12 ± (√33)/6

What substitution should be used to rewrite 6(x + 5)² + 5(x + 5) - 4 = 0 as a quadratic equation?

Summary:

To convert the above equation into a quadratic equation x + 5 should be substituted by u and rewritten as 6u² + 5u - 4 = 0