91Ó°ÊÓ

The substitution method is a powerful tool for solving equations. It involves replacing a part of the equation with a new variable to simplify the process.

In the given problem, we start with the equation: $$ 2(x+3)^{2}=5(x+3)-2 $$
Here, we define a new variable:
$$ u = x + 3 $$
Substituting this into the equation transforms it into: $$ 2u^2 = 5u - 2 $$
This substitution reduces the complexity of the equation, making it easier to handle.

Why use substitution?

• Simplifies complex expressions.
• Makes equations easier to solve.
• Helps in isolating variables more efficiently

Once simplified to a familiar form, solving becomes straightforward. After solving for $$ u $$, we back-substitute to find the original variable. This method is especially useful in solving quadratic equations.

When faced with a quadratic equation in the form $$ au^2 + bu + c = 0 $$, the quadratic formula is a reliable method to find solutions.

The quadratic formula is: $$ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
This formula provides the solutions for the quadratic equation, allowing us to determine the values of the unknown variable.
Let's apply this to the equation we obtained from our substitution: $$ 2u^2 - 5u + 2 = 0 $$
Here,

• a = 2
• b = -5
• c = 2

Plug these values into the quadratic formula to get:
$$ u = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} $$
Simplifying further, we calculate:
$$ u = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm 3}{4} $$
This provides the solutions for $$ u $$:

• $$ u = \frac{5 + 3}{4} = 2 $$
• $$ u = \frac{5 - 3}{4} = \frac{1}{2} $$

Using the quadratic formula ensures accurate and efficient solutions.

The discriminant in a quadratic equation is an important value that gives insights into the nature of the roots of the equation.

The discriminant is given by: $$ \Delta = b^2 - 4ac $$
Based on its value, we can determine the nature of the roots:

• $$ \Delta > 0 $$: Two distinct real roots.
• $$ \Delta = 0 $$: One real root (repeated).
• $$ \Delta < 0 $$: No real roots (complex roots).

In our problem, substituting the values: $$ a = 2 $$, $$ b = -5 $$, and $$ c = 2 $$, we calculate:
$$ \Delta = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9 $$
Since $$ \Delta = 9 > 0 $$, there are two distinct real roots. These roots are:

• $$ u = 2 $$
• $$ u = \frac{1}{2} $$

Understanding the discriminant helps anticipate the nature of the solutions before solving the equation.