Mastering Variable C: Solving For Success With Step-By-Step Guidance

To solve for c in an equation: isolate it by subtracting b from both sides. Divide both sides by the coefficient a (if non-zero) to solve directly. If factorization is possible, set each factor equal to zero to find c. For quadratic equations, use the quadratic formula: c = (-b ± sqrt(b² – 4ac)) / 2a. The discriminant (b² – 4ac) determines the number and nature of roots.

Isolating Terms: Subtracting b from Both Sides

When faced with an equation that has a term containing a variable on both sides, our goal is to isolate that variable on one side of the equation. One of the most fundamental techniques for achieving this is subtracting a constant term from both sides.

Let’s say we have an equation like:

c + b = d

To isolate the term containing c, we want to get rid of the term b on the left side. We do this by subtracting b from both sides of the equation:

c + b - b = d - b

This operation keeps the equation balanced, as we are performing the same action on both sides. By subtracting b from both sides, we essentially cancel out the b term on the left side, leaving us with:

c = d - b

This new equation now has the term containing c isolated on the left side, making it much easier to solve for c by simply performing basic arithmetic.

Solving for a Variable: Dividing Both Sides by the Coefficient

In the realm of algebra, solving equations is akin to an exciting treasure hunt, where we aim to uncover the hidden value of a variable. One of the most fundamental techniques in this quest is dividing both sides of an equation by a non-zero coefficient.

Imagine you have an equation like this: 2x + 5 = 13. Can you tell me the value of x? Intuition might tell you to subtract 5 from both sides, but that’s just the first step. To isolate x, we need to get rid of the 2 in front of it.

This is where the magic of dividing by a coefficient comes in. We can divide both sides of the equation by 2, provided it’s not zero. Why is this such a big deal? Because division is like the inverse operation of multiplication. If you multiply something by a number and then divide it by the same number, you get back what you started with.

So, let’s divide both sides of our equation by 2 and see what happens:

(2x + 5) / 2 = 13 / 2

Now, the 2 on the left-hand side cancels out, and we’re left with:

x + 5/2 = 13/2

Voilà! We have isolated the variable x. To find its value, we can subtract 5/2 from both sides:

x = 13/2 - 5/2

x = 4

And there you have it! By dividing both sides of the equation by the coefficient of the variable (2 in this case), we were able to solve for x. This technique is a cornerstone of algebra, and it will serve you well in your mathematical adventures.

Unveiling the Secrets of Factoring: A Journey to Solve Quadratic Equations

In the realm of algebra, quadratic equations hold a prominent place, often leaving aspiring learners feeling perplexed. But fear not, dear reader, for today we embark on a captivating journey, exploring the technique of factoring to unravel these enigmatic equations.

Imagine you’ve encountered a quadratic equation like 2x² – 5x + 2 = 0. To solve this puzzle, we’ll employ the remarkable skill of factoring. This technique seeks to express the equation as a product of two or more simpler expressions.

Let’s start by identifying common factors that appear in each term. In our example, both 2x and 2 appear in all three terms. We can extract this common factor as follows:

2x² - 5x + 2 = (2x)(x) - (2x)(2/2) + (2)(1)

Now, observe that we have a difference of squares in the first two terms: (2x)(x) – (2x)(2). This can be factored further using the formula (a – b)(a + b) = a² – b². Applying this to our equation, we get:

2x² - 5x + 2 = (2x - 2)(x + 1)

Now, we have two factors that are each equal to zero when their contents are zero. So, we set each factor equal to zero:

(2x - 2) = 0 or (x + 1) = 0

Solving for x in each equation, we find:

x = 1 or x = -1/2

These values of x represent the solutions to our original quadratic equation. By factoring the equation and setting each factor equal to zero, we have effectively broken down the problem into smaller, more manageable pieces, ultimately revealing its secrets.

Unveiling the Power of the Quadratic Formula

In the realm of mathematics, quadratic equations are like enigmatic puzzles, hiding their solutions beneath a veil of algebra. Fear not, for we have a secret weapon in our arsenal: the quadratic formula. This magical formula is a key that unlocks the secrets of these quadratic puzzles.

Understanding the Quadratic Formula:

The quadratic formula is a mathematical tool designed to solve quadratic equations of the form ax² + bx + c = 0. It takes the form of:

x = (-b ± √(b² - 4ac)) / 2a

where:

• a, b, and c are the coefficients of the quadratic equation.
• ± represents the “plus or minus” sign.
• √ is the square root symbol.

Step-by-Step Breakdown:

Step 1: Plug in the Values

Substitute the values of a, b, and c into the formula.

Step 2: Calculate the Discriminant

The discriminant is the expression b² – 4ac. It determines the nature of the roots.

Step 3: Find the Roots

Use the formula to calculate two possible values for x, one with the (+) sign and the other with the (-) sign.

Interpreting the Discriminant:

The discriminant holds the key to understanding the behavior of the equation:

• Positive Discriminant: Two distinct real solutions.
• Zero Discriminant: One repeated real solution (double root).
• Negative Discriminant: No real solutions (complex roots).

Example:

Let’s solve the equation x² – 5x + 6 = 0 using the quadratic formula:

• a = 1, b = -5, c = 6
• Discriminant: (-5)² – 4(1)(6) = 25 – 24 = 1
• x = (-(-5) ± √1) / 2(1)
• x = (5 ± 1) / 2
• x = 3 or x = 2

The solutions to this equation are x = 3 and x = 2, which are both real and distinct.