Question

Find the real solutions of each equation. $$ (2 x+5)^{2}-(2 x+5)-6=0 $$

Short answer

The real solutions are \( x = -1 \) and \( x = -\frac{7}{2} \).

Step-by-step solution

- Substitute a Variable

Let \( u = 2x + 5 \). This simplifies the equation. So it becomes: \( u^2 - u - 6 = 0 \).

- Factor the Quadratic

Factor the equation \( u^2 - u - 6 = 0 \) to find its roots. The factors are \( (u-3)(u+2) = 0 \).

- Solve for u

Set each factor equal to zero and solve for \( u \). \( u - 3 = 0 \) gives \( u = 3 \). \( u + 2 = 0 \) gives \( u = -2 \).

- Substitute Back x

Replace \( u \) with \( 2x + 5 \) in each solution found for \( u \). For \( u = 3 \): \( 2x + 5 = 3 \). Solve for \( x \): \( 2x = -2 \) \( x = -1 \). For \( u = -2 \): \( 2x + 5 = -2 \). Solve for \( x \): \( 2x = -7 \) \( x = -\frac{7}{2} \).

- Write the Final Solutions

The real solutions to the equation are \( x = -1 \) and \( x = -\frac{7}{2} \).

Key concepts

substitution method

The substitution method is an effective strategy for simplifying complex equations. When given a cumbersome expression, we can replace part of it with a new variable. This makes the equation easier to manage.

For instance, in the given equation: \[ (2x+5)^{2}-(2x+5)-6=0 \] we substitute \[ u = 2x + 5 \] to simplify the equation to \[ u^2 - u - 6 = 0 \].

This substitution transforms the original equation into a quadratic form. By simplifying in this way, solving the equation becomes more straightforward, as it reduces the complexity.

factoring quadratics

Factoring quadratics is a method to solve equations of the form \[ ax^2 + bx + c = 0 \]. The goal is to express the quadratic equation as a product of two binomials.

For the simplified equation \[ u^2 - u - 6 = 0 \], we look for two numbers that multiply to \[ -6 \] (the constant term) and add to \[ -1 \] (the coefficient of \[ u \]).
These numbers are \[ -3 \] and \[ 2 \]
This allows us to write the equation as: \[ (u - 3)(u + 2) = 0 \].

Factoring makes it easy to see the solutions to the equation, as each factor can be set to zero individually.

real solutions

Real solutions are the values of the variable that satisfy the equation within the set of real numbers. To find these, we solve for the variable after factoring.

From the factored form \[ (u - 3)(u + 2) = 0 \], we set each factor to zero to find: \[ u - 3 = 0 \] and \[ u + 2 = 0 \].
Solving these gives the real solutions \[ u = 3 \] and \[ u = -2 \].

Since we initially substituted \[ u = 2x + 5 \], we substitute back into each solution:

• For \[ u = 3 \]: Solve \[ 2x + 5 = 3 \] giving \[ x = -1 \]
• For \[ u = -2 \]: Solve \[ 2x + 5 = -2 \] giving \[ x = -\frac{7}{2} \]

These solutions show where the original equation equals zero.

solving for variables

Solving for variables is the process of isolating the variable on one side of the equation. For the given problem, once we find the roots for \[ u \], we substitute back to the original variable \[ x \] to find its values.

For \[ u = 3 \]:

• Replace \[ u \] with \[ 2x + 5 \]
• Set \[ 2x + 5 = 3 \]
• Solve \[ 2x = -2 \]
• Then, \[ x = -1 \]

For \[ u = -2 \]:

• Replace \[ u \] with \[ 2x + 5 \]
• Set \[ 2x + 5 = -2 \]
• Solve \[ 2x = -7 \]
• Then, \[ x = -\frac{7}{2} \]

Therefore, the final solutions for the variable \[ x \] are \[ x = -1 \] and \[ x = -\frac{7}{2} \].