Problem 66 - Algebra A Combined Function

Question

Solve each equation. $$ 2 x^{2}+12 x-1=4+3 x $$

Step-by-Step Solution

Answer

Solutions are \( x = \frac{1}{2} \) and \( x = -5 \).

Step 1: Set the equation to zero

Start by moving all terms to one side of the equation by subtracting \(4+3x\) from both sides. This gives: \[ 2x^2 + 12x - 1 - 4 - 3x = 0 \] Simplify this to: \[ 2x^2 + 9x - 5 = 0 \]

Step 2: Use the quadratic formula

Identify the coefficients from the simplified quadratic equation \(2x^2 + 9x - 5 = 0\). Here, \(a = 2\), \(b = 9\), and \(c = -5\). Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values into the formula: \[ x = \frac{-9 \pm \sqrt{9^2 - 4(2)(-5)}}{2(2)} \] This simplifies to: \[ x = \frac{-9 \pm \sqrt{81 + 40}}{4} \]

Step 3: Simplify the expression under the square root

Calculate the discriminant in the quadratic formula expression. Simplify inside the square root: \[ x = \frac{-9 \pm \sqrt{121}}{4} \] Since \(\sqrt{121} = 11\), substitute back to get: \[ x = \frac{-9 \pm 11}{4} \]

Step 4: Solve for each possible value of x

Calculate the two potential solutions for \(x\):- For \(x = \frac{-9 + 11}{4}\): \[ x = \frac{2}{4} = \frac{1}{2} \]- For \(x = \frac{-9 - 11}{4}\): \[ x = \frac{-20}{4} = -5 \]

Step 5: Conclude the solutions

The solutions to the equation \( 2x^2 + 12x - 1 = 4 + 3x \) are \( x = \frac{1}{2} \) and \( x = -5 \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is especially useful when the equation cannot be easily factored. The quadratic formula is represented as:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

When applied, this formula provides the solutions for any quadratic equation of the form \( ax^2+bx+c = 0 \). By substituting known values of \( a \), \( b \), and \( c \) into the formula, you can find the values of \( x \) (the variable) that satisfy the equation. In a quadratic equation, \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.

Discriminant

The discriminant is a key part of the quadratic formula found under the square root symbol: \( b^2 - 4ac \). Its value determines the nature of the roots for a quadratic equation:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution (a repeated or double root).
• If the discriminant is negative, there are no real solutions, but two complex solutions.

Discriminant helps predict the type of solutions without solving the equation. In the exercise, the discriminant is \( 121 \), a positive number, indicating that there are two distinct real solutions: \( x = \frac{1}{2} \) and \( x = -5 \).

Simplifying Expressions

Simplifying expressions is an essential part of solving equations, as it makes equations easier to handle and solve. In the original exercise, you first rearrange the terms:

• Subtract \( 4 + 3x \) from both sides, resulting in: \( 2x^2 + 12x - 1 - 4 - 3x = 0 \).
• Combine like terms to simplify: \( 2x^2 + 9x - 5 = 0 \).

Combining like terms is crucial for reducing the equation to a simpler form, paving the way to apply the quadratic formula effectively.

Solving Equations

Solving equations is the process of finding the variable's values that satisfy a given equation. For quadratic equations, this process typically involves several steps:

• Set the equation to zero: Move all terms to one side, as shown in the exercise, so that the quadratic formula can be applied.
• Identify coefficients: Recognize values of \( a \), \( b \), and \( c \) from the equation.
• Apply the quadratic formula: Substitute the coefficients into the formula and simplify.
• Calculate possible solutions: Determine the solutions by evaluating the simplified equation.

In this exercise, these steps help arrive at the solutions: \( x = \frac{1}{2} \) and \( x = -5 \), thoroughly solving the quadratic equation.