Problem 18 - Beginning and Intermediate Algebra

Question

Solve using the zero-factor property. $$ 5 x^{2}-14 x=3 $$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{1}{5}\) and \(x = 3\).

Step 1: Move all terms to one side

Subtract 3 from both sides of the equation to set it equal to zero: \[5x^2 - 14x - 3 = 0\]

Step 2: Factor the quadratic equation

Factor the quadratic equation if possible. We look for two numbers that multiply to \(5 * -3 = -15\) and add to \(-14\). These numbers are \(-15\) and \(+1\). Rewrite the middle term using these numbers: \[5x^2 - 15x + 1x - 3 = 0\]

Step 3: Group and factor by grouping

Group the terms into two pairs and factor each pair: \[(5x^2 - 15x) + (1x - 3) = 0\] Factor out the greatest common factor from each group: \[5x(x - 3) + 1(x - 3) = 0\]

Step 4: Factor out the common binomial

Factor out the common binomial factor: \[(5x + 1)(x - 3) = 0\]

Step 5: Apply the zero-factor property

Set each factor equal to zero and solve for x: \[5x + 1 = 0\] Solve for x: \[x = -\frac{1}{5}\] \[x - 3 = 0\] Solve for x: \[x = 3\]

Key Concepts

quadratic equations

Quadratic equations are mathematical expressions that involve terms up to the second degree (squared terms). They take the general form: ax^2 + bx + c = 0where 'a', 'b', and 'c' are constants and 'x' represents the variable. To solve quadratic equations, we often use several methods:

• Factoring
• The Quadratic Formula
• Completing the Square

In this exercise, we solve the quadratic equation by factoring and applying the zero-factor property. Factoring is an efficient technique to break down the equation into simpler components. From there, each component (or factor) is set to zero to find the possible values for 'x'. Using the zero-factor property, where if the product of two factors is zero, at least one of the factors must be zero.

factoring

Factoring is a mathematical process where equations are broken down into simpler expressions called factors. For a quadratic equation in the form ax^2 + bx + c = 0, our goal is to rewrite it as two binomials: (px + q)(rx + s) = 0. To reach this form:

• First, subtract or add constants to set the equation to zero, ensuring it is in standard form.
• Next, identify two numbers that multiply to the product of 'a' and 'c' and add to 'b'. In our example, these numbers are -15 and +1.
• Rewrite the middle term using these new numbers, forming a four-term polynomial.
• Group the terms and factor out the greatest common factor from each group.
• Finally, factor out the common binomial.

Following these steps, the equation 5x^2 - 14x - 3 was rewritten and factored as (5x + 1)(x - 3) = 0.

solving algebraic equations

Solving algebraic equations involves finding the value(s) of the variable(s) that make the equation true. For a factored quadratic equation like (5x + 1)(x - 3) = 0, we use the zero-factor property. This property states that if a product of factors is zero, then at least one of the factors must be zero.

• Set each factor equal to zero independently.
• Solve for 'x' in each simple equation.

For example, we solve:

• 5x + 1 = 0, leading to x = -1/5
• x - 3 = 0, leading to x = 3

Therefore, the solutions to the original equation 5x^2 - 14x = 3 are x = -1/5 and x = 3. These solutions make the initial equation true when substituted back, confirming our solutions.