Problem 66 - Elementary Algebra

Question

Solve. $$ 2 x x+5-12 x-3=4-7 x 2 \times 2+7 x-15 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{19 + \sqrt{33}}{4}\) and \(x = \frac{19 - \sqrt{33}}{4}\).

Step 1: Simplify the Equation

Let's first simplify both sides of the equation by resolving any mixed operations and combining like terms. The given equation is \(2x^2 + 5 - 12x - 3 = 4 - 7 \cdot 2 \times 2 + 7x - 15\).First, let's simplify both sides:- On the left side: \(2x^2 + 5 - 12x - 3 = 2x^2 - 12x + 2\).- On the right side: \(4 - 7 \cdot 2 \times 2 + 7x - 15 = 4 - 28 + 7x - 15 = 7x - 39\).So, the equation simplifies to: \(2x^2 - 12x + 2 = 7x - 39\).

Step 2: Combine Like Terms on One Side

To solve for \(x\), we want to collect all the \(x\)-terms on one side and the constant terms on the other. We start with:\[2x^2 - 12x + 2 = 7x - 39\]Subtract \(7x\) and add \(39\) on both sides:\[2x^2 - 12x - 7x + 2 + 39 = 0\]Simplify:\[2x^2 - 19x + 41 = 0\].

Step 3: Apply the Quadratic Formula

Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -19\), and \(c = 41\). First, calculate the discriminant:\[b^2 - 4ac = (-19)^2 - 4 \cdot 2 \cdot 41 = 361 - 328 = 33\].Since the discriminant is positive, there are two real solutions.

Step 4: Solve for x Using the Quadratic Formula

Substitute the values and compute the roots:\[x = \frac{-(-19) \pm \sqrt{33}}{2 \cdot 2} = \frac{19 \pm \sqrt{33}}{4}\]This gives us:1. \(x = \frac{19 + \sqrt{33}}{4}\)2. \(x = \frac{19 - \sqrt{33}}{4}\)

Key Concepts

Simplifying Expressions

To make solving quadratic equations easier, we first need to simplify expressions. This involves resolving mixed operations and combining like terms. Simplifying makes the equation clearer and easier to handle.

In the original exercise, the equation is:

• On the left side: Combine and simplify terms like \(2x^2 - 12x + 5 - 3\), which results in \(2x^2 - 12x + 2\).
• On the right side: Simplify \(4 - 7 \cdot 2 \times 2 + 7x - 15\) to get \(7x - 39\).

Step by step, simplify each part of the expression.
This process reduces complex equations into manageable forms.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of quadratic equations of the form \(ax^2 + bx + c = 0\). It's expressed as:

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

This formula works universally for any quadratic equation.

In our example, \(a = 2\), \(b = -19\), and \(c = 41\).
Before jumping into the solution, calculate the discriminant \(b^2 - 4ac\):

• \((-19)^2 - 4 \cdot 2 \cdot 41 = 361 - 328 = 33\)

Since the discriminant is positive (33), there are two real solutions to the equation.
This tells us the quadratic equation has two distinct roots.

Solving Equations

Solving quadratic equations means finding the values of \(x\) that satisfy the equation. After simplifying, the equation becomes easier to solve. We use the quadratic formula as explained.

Substitute the values into the formula:

• First root: \(x = \frac{19 + \sqrt{33}}{4}\)
• Second root: \(x = \frac{19 - \sqrt{33}}{4}\)

Each solution gives a possible value for \(x\) that makes the original equation true.
This approach ensures you find all possible solutions in a systematic way.