Problem 36 - Algebra 2

Question

Solve each equation by factoring. \(4 x^{2}-17 x=-4\)

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = \frac{1}{4}\).

Step 1: Write the equation in standard form

Move all terms to one side of the equation to set it equal to zero. Start with the given equation: \[4x^2 - 17x = -4\]Add 4 to both sides to obtain:\[4x^2 - 17x + 4 = 0\]

Step 2: Identify a, b, and c

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). From \(4x^2 - 17x + 4 = 0\), identify:\(a = 4\), \(b = -17\), and \(c = 4\).

Step 3: Factor the quadratic equation

Look for two numbers that multiply to \(a imes c = 4 imes 4 = 16\) and add to \(b = -17\). These numbers are \(-16\) and \(-1\) because \(-16 imes -1 = 16\) and \(-16 + (-1) = -17\).

Step 4: Write the equation with the factors

Rewrite the middle term \(-17x\) using the pair found:\[4x^2 - 16x - x + 4 = 0\]

Step 5: Factor by grouping

Group the terms to factor by grouping:\[(4x^2 - 16x) + (-x + 4) = 0\]Factor the first group and the second group:\[4x(x - 4) - 1(x - 4) = 0\]Now factor out the common factor \((x - 4)\):\[(x - 4)(4x - 1) = 0\]

Step 6: Solve each factor for x

Set each factor equal to zero and solve for \(x\):1. \(x - 4 = 0\) \(x = 4\) 2. \(4x - 1 = 0\) \(4x = 1\) \(x = \frac{1}{4}\)

Step 7: Present the solutions

The solutions to the equation \(4x^2 - 17x + 4 = 0\) are \(x = 4\) and \(x = \frac{1}{4}\).

Key Concepts

Factoring

Factoring is a method used to express a polynomial as a product of simpler polynomials. In quadratic equations, this technique helps break down complex expressions into two binomials. The goal is to discover numbers that multiply to a certain product while also adding to a particular sum.

To factor a quadratic equation, like in the given exercise, follow these steps:

• Identify the constant values of the equation: the coefficient in front of the square term (\( a \)), the linear term (\( b \)), and the constant term (\( c \)).
• Look for two numbers that multiply to \( a \times c \) and add to \( b \).

Once these numbers are found, the quadratic can be rewritten to reflect these values and eventually factored by grouping, shedding light on the possible solutions.

Solving Equations

Solving equations involves finding the values of the variables that satisfy the equation. Quadratic equations, like the one in our problem, are solved by finding solutions that make the entire expression equal zero. There are multiple ways to solve quadratic equations such as factoring, completing the square, and using the quadratic formula.

In this case, we applied the factoring method, which involves rewriting the quadratic in a form that highlights the possible values of \( x \).

• Set each factor equal to zero. This reveals the candidate solutions for the equation.
• By solving these smaller, simpler equations, we find all potential solutions for the quadratic.

In our exercise, the solutions denote the special values of \( x \) where the expression's polynomial result balances at zero.

Polynomials

Polynomials are mathematical expressions that include terms composed of variables and coefficients. They can take various forms, from simple linear polynomials to more complex quadratic and cubic polynomials.

The quadratic polynomials, like \( 4x^2 - 17x + 4 \) in our exercise, consist of three terms:

• Squaring term (\( a x^2 \)) – affects the width and orientation of the parabola when graphed.
• Linear term (\( bx \)) – shifts the parabola left or right.
• Constant term (\( c \)) – moves the parabola up or down.

Understanding the structure of polynomials allows manipulation into different forms and solving for variable values effectively. These expressions are foundational to algebra and calculus. Factoring and solving them are crucial for deeper mathematical studies.