Question

Solve each equation, if possible. $$ x(2 x-3)=(2 x+1)(x-4) $$

Short answer

x = -1

Step-by-step solution

- Expand Both Sides

Expand both sides of the equation by using the distributive property. On the left side, distribute x through the binomial. On the right side, use the FOIL method to expand the product of the two binomials. This gives: Left Side: \[ x(2x - 3) = 2x^2 - 3x \] Right Side: \[ (2x + 1)(x - 4) = 2x^2 - 8x + x - 4 = 2x^2 - 7x - 4 \]

- Set the Equations Equal

Set the expanded forms equal to each other:\[2x^2 - 3x = 2x^2 - 7x - 4 \]

- Subtract Both Sides

Subtract \(2x^2\) from both sides to eliminate the quadratic term: \[ -3x = -7x - 4 \]

- Isolate the Variable

Add \(7x\) to both sides to combine like terms and isolate the variable:\[4x = -4 \]

- Solve for x

Divide both sides of the equation by 4 to solve for \(x\):\[x = -1 \]

- Verify the Solution

Substitute \(x = -1\) back into the original equation to verify the solution:Original Equation: \[ x(2x - 3) = (2x + 1)(x - 4) \]Substitute \(x = -1\):\[ -1(2(-1) - 3) = (2(-1) + 1)(-1 - 4) \]Simplify:\[ -1(-2 - 3) = (-2 + 1)(-5) \]\[ -1(-5) = (-1)(-5) \]\[ 5 = 5 \]Since both sides of the equation are equal, \(x = -1\) is the correct solution.

Key concepts

Distributive Property

The distributive property is a fundamental algebraic principle. It allows us to multiply a single term by every term inside a parenthesis. In our exercise, we used this property on the left side of the equation involving the term \( x \(2x - 3\)\)) to obtain \( 2x^2 - 3x \). Here's how it works step-by-step:

• Multiply \( x \cdot 2x \) to get \( 2x^2 \).
• Then, multiply \( x \cdot (-3) \) to get \( -3x \).

By distributing the term \( x \) across each term in the parenthesis, we explicitly outline how each component contributes to the expanded form.

FOIL Method

The FOIL method is a special case of the distributive property used primarily for multiplying two binomials. FOIL stands for First, Outer, Inner, and Last—highlighting which terms to multiply. In our problem, the right side uses the FOIL method on \( (2x + 1)(x - 4) \). Here’s the detailed breakdown:

• First terms: \( 2x \cdot x = 2x^2 \).
• Outer terms: \( 2x \cdot (-4) = -8x \).
• Inner terms: \( 1 \cdot x = x \).
• Last terms: \( 1 \cdot (-4) = -4 \).

Combining these products, we get the expanded form \( 2x^2 - 8x + x - 4 = 2x^2 - 7x - 4 \). The FOIL method simplifies multiplying binomials and ensures no term is missed.

Isolate Variable

Isolating the variable means arranging the equation so that the variable (typically x) is alone on one side of the equation. This helps find its value. Starting from: \(2x^2 - 3x = 2x^2 - 7x - 4 \), we first subtract \(2x^2 \) from both sides to simplify:
\[ -3x = -7x - 4 \]
Then, we move the terms involving the variable by adding \(7x \) to both sides, giving:
\[ 4x = -4 \]
This process gathers all variable terms together and constants on the other side, making it easier to solve for x. Finally, by dividing both sides by 4, we isolate the variable: \[ x = -1 \].

Verify Solution

Verifying the solution ensures the value found satisfies the original equation. We substitute \( x = -1 \) back into the equation \( x(2x - 3) = (2x + 1)(x - 4) \):

• Left Side: \[ -1(2(-1) - 3) = -1(-5) = 5 \].
• Right Side: \[ (2(-1) + 1)(-1 - 4) = (-2 + 1)(-5) = (-1)(-5) = 5 \].

Since both sides equal 5, the solution \( x = -1 \) is correct. Verifying ensures the answer correctly resolves the original equation, confirming no steps were overlooked.