Problem 33 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ (2 x+1)^{2}=x+4 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{-3 + \sqrt{57}}{8} \) and \( x = \frac{-3 - \sqrt{57}}{8} \).

Step 1: Expand the left-hand side

Expand \((2 x + 1)^{2}\) to get the equation in standard form. \[ (2x + 1)^{2} = 2x + 1 \times 2x + 1 = 4x^2 + 4x + 1 \]

Step 2: Set up the equation in standard quadratic form

Subtract \(x + 4\) from both sides to set the equation to zero: \[ 4x^2 + 4x + 1 - (x + 4) = 0 \] Simplify to \[ 4x^2 + 3x - 3 = 0 \]

Step 3: Identify coefficients

In the standard form \(ax^2 + bx + c = 0\): \[ a = 4 \] \[ b = 3 \] \[ c = -3 \]

Step 4: Apply the quadratic formula

Use the quadratic formula, \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], to find the roots. \[ x = \frac{-3 \pm \sqrt{3^2 - 4(4)(-3)}}{2(4)} = \frac{-3 \pm \sqrt{9 + 48}}{8} \]

Step 5: Simplify under the square root

Simplify \sqrt{9 + 48}\: \[ \sqrt{57} \] So, the formula becomes: \[ x = \frac{-3 \pm \sqrt{57}}{8} \]

Step 6: Solve for the roots

The solutions to the equation are: \[ x = \frac{-3 + \sqrt{57}}{8}, \quad x = \frac{-3 - \sqrt{57}}{8} \]

Key Concepts

quadratic equation

A quadratic equation is a second-degree polynomial equation in a single variable. The general form of a quadratic equation is expressed as:
ax^2 + bx + c = 0, where
a, b, and c are constants, and 'a' cannot be zero.
Quadratic equations are essential in algebra because their solutions, or 'roots', determine where the equation equals zero. This can be visualized as the points where a parabolic graph intersects the x-axis.

solving equations

Solving equations involves finding the variable's value that makes the equation true. For quadratic equations, one efficient method is the quadratic formula.
To solve a quadratic equation using the quadratic formula, you follow these steps:

• 1. Write the equation in the standard form: ax^2 + bx + c = 0.


• 2. Identify the coefficients a, b, and c.


• 3. Plug these coefficients into the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \],


• 4. Simplify the expression under the square root (discriminant).


• 5. Compute the values for 'x' by including the positive and negative roots.

This method provides solutions for the roots of the quadratic equation. It's a powerful tool for solving many algebra problems.

algebra simplification

Simplifying algebraic expressions is a key step in solving equations. It helps in making complex expressions easier to work with.
Here are some essential simplification techniques used in our solution:

• Expanding: Multiplying out expressions, like (2x + 1)^2, to write them in a simplified form such as 4x^2 + 4x + 1.


• Combining like terms: Simplifying similar terms on both sides of the equation. For example, simplifying 4x^2 + 4x + 1 - (x + 4) to get 4x^2 + 3x - 3.

Mastery of these techniques is vital for solving more complex equations efficiently.

roots of an equation

The roots of an equation are the values of the variable that satisfy the equation. For quadratic equations, these are the 'x' values where the quadratic function hits the x-axis.
In our example, we used the quadratic formula to find the roots: \( x = \frac{-3 \pm \sqrt{57}}{8} \).
So we got two roots:

• \( x = \frac{-3 + \sqrt{57}}{8} \)


• \( x = \frac{-3 - \sqrt{57}}{8} \).

These solutions tell us the points where the original quadratic equation equals zero. Understanding roots is important for analyzing the behavior of quadratic functions.