How to find the solution to x2 – 10x = 24

Solving the Equation

To solve the equation:

We start by moving all terms to one side to set the equation to zero:

This is a quadratic equation of the form $ax^2 + bx + c = 0$, where:

Factoring the Quadratic Equation

We look for two numbers that multiply to $ac = -24$ and add to $b = -10$.

The numbers that satisfy these conditions are -12 and 2 because:

Thus, we can factor the quadratic as follows:

Solving for x

To find the values of $x$, we set each factor to zero:

So, solving these equations separately:

Conclusion

The solutions to the equation are x = 12 and x = -2.

Understanding the Best Way to Solve the Given Equation

When tasked with solving a given equation, it’s important to follow a structured approach to ensure accuracy and efficiency. Here’s a walkthrough of essential steps and considerations:

Step-by-Step Approach

1. Identify the Type of Equation

   • Determine whether you are dealing with a linear, quadratic, polynomial, exponential, logarithmic, or another type of equation. The methods used to solve them can vary significantly.

2. Simplify the Equation

   • Combine like terms and simplify both sides of the equation as much as possible. This might involve expanding expressions, factoring, or reducing fractions.

3. Isolate the Variable

   • For simpler equations, your goal is to get the variable you are solving for on one side of the equation. This may involve adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

4. Special Techniques for Different Equations

   • For quadratic equations, use factoring, completing the square, or the quadratic formula: $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$$
   • For polynomial equations, factoring can be combined with the Rational Root Theorem.
   • For exponential equations, take the logarithm of both sides. For example, for $e^x = y$, apply $\ln$: $$\ln(e^x) = \ln(y) \implies x = \ln(y)$$

5. Check for Extraneous Solutions

   • Especially when dealing with higher-order polynomials, trigonometric, and logarithmic equations, it’s possible to generate extraneous solutions during manipulation. Always substitute back into the original equation to verify solutions.

6. Graphical Method

   • Occasionally, it’s useful to graph both sides of the equation and look for points of intersection. This visual approach can provide insights or confirm analytical solutions.

Practical Example

Consider the quadratic equation:

1. Identify and Simplify: Recognize it as a quadratic equation and simplify if necessary (none required here).

2. Use the Quadratic Formula: Given $a=2, b=-4, c=-6$, apply the formula:

   $$x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4 \cdot 2 \cdot (-6)}}}}{2 \cdot 2}$$ $$x = \frac{{4 \pm \sqrt{{16 + 48}}}}{4}$$ $$x = \frac{{4 \pm \sqrt{64}}}{4}$$ $$x = \frac{{4 \pm 8}}{4}$$

3. Solve for x:

   $$x = \frac{12}{4} = 3 \quad \text{or} \quad x = \frac{-4}{4} = -1$$

4. Verify: Substitute $x = 3$ and $x = -1$ back into the original equation to ensure they are solutions.

Conclusion

Solving equations requires a systematic approach tailored to the equation type. The steps outlined above should provide a clear framework for tackling a variety of equations. Feel free to adapt these techniques depending on the complexity and nature of the given equation.

Understanding the Quadratic Formula and Factoring

When dealing with quadratic equations, two fundamental methods you often encounter are the quadratic formula and factoring. Each method provides a way to find the solutions, or "roots", of a quadratic equation of the form:

Quadratic Formula

The quadratic formula is a straightforward way to solve any quadratic equation. The formula is:

Here's a breakdown of each component:

• $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.
• $b^2 - 4ac$ is called the discriminant, and it determines the nature of the roots.
  • If it's positive, there are two distinct real roots.
  • If it's zero, there is exactly one real root (also called a repeated or double root).
  • If it's negative, there are two complex roots.

Factoring

Factoring involves rewriting the quadratic equation as a product of its linear factors, if they exist. Consider a quadratic equation in its simplified form:

To factor it, we assume it can be written as:

Here, $m$, $n$, $p$, and $q$ are numbers that satisfy the condition $mp = a$, $nq = c$, and $mq + np = b$.

If you can identify such numbers, the equation factors into two binomials:

Setting each factor to zero gives the solutions:

Example:

Consider the quadratic equation:

Using the Quadratic Formula: Here, $a = 2$, $b = -4$, and $c = -6$. Plugging these into the quadratic formula, we get:

Thus, the solutions are:

Using Factoring: First, factor the equation:

Divide through by 2:

Next, find two numbers that multiply to $-3$ and add to $-2$:

Setting each factor to zero:

In summary: Both methods confirm the roots are $x = 3$ and $x = -1$. The quadratic formula works universally, while factoring relies on the quadratic being factorable into integers.