What are the solutions of the equation (2x + 3)^2 + 8(2x + 3) + 11 = 0? Use u substitution and the quadratic formula to solve.
Mathematics · Middle School · Thu Feb 04 2021
Answered on
Given the equation:
(2x + 3)^2 + 8(2x + 3) + 11
Determine the solution.
Solution:
Let u = 2x + 3
Subsitute u to the value of 2x + 3.
= (2x + 3)^2 + 8(2x + 3) + 11
=u^2 + 8u + 11
a = 1
b = 8
c = 11
Solve using the quadratic formula.
The Quadratic formula:
x = −b ± √(b^2 − 4ac)/2a
is used to solve quadratic equations where a ≠ 0, in the form
ax^2+bx+c=0
When b^2−4ac=0 there is one real root.
When b^2−4ac>0 there are two real roots.
When b^2−4ac<0 there are no real roots, only a complex number.
Substitute the given values of a, b and c to the quadratic formula.
u = −b ± √(b^2 − 4ac)/2a
u = −(8) ± √((8)^2 − 4(1)(11))/2(1)
u = −8 ± √(64 - 44)/2
u = −8 ± √(20)/2
u = −8 ± 4.47/2
Solve for ± separately.
u = -8 + 4.47 / 2
u = -4.47/2
u = -2.235
u = -8 - 4.47/2
u = -12.47/2
u = -6.235
Final answer:
u = -2.235
u = -6.235