Problem 14 - Algebra 1: Concepts and Skills

Question

Write the quadratic equation in standard form. $$-x^{2}=15$$

Step-by-Step Solution

Answer

The quadratic equation in standard form is \(x^{2} - 15 = 0\)

Step 1: Reversing the Sign

To begin, the standard form has the left-hand side as positive, so we need to change the negative sign at the start of the equation. Multiply both sides of the equation by -1, giving the equation \(x^{2} = -15\)

Step 2: Moving the Constant

Now, we need to set the quadratic equation equal to zero. This can be done by moving the constant on the right side of the equation to the left. Subtract 15 from both sides of the equation to get \(x^{2} - 15 = 0\)

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and primarily involve a single variable raised to the power of two, typically expressed in the form \( ax^2 + bx + c = 0 \). These equations appear in many real-world scenarios, such as projectile motion, area calculations, and even economics. The coefficients \(a\), \(b\), and \(c\) are usually real numbers, and the first term, \(ax^2\), defines the degree of the equation as quadratic.
Depending on the type of coefficients, every quadratic equation can have zero, one, or two solutions in the realm of real numbers. The solutions to these equations, called roots, can be found using different methods such as factoring, completing the square, or using the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Whether on paper or through mathematical intuition, understanding quadratic equations is critical for solving more complex algebraic problems.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to reach a solution or standard form. In our exercise, the goal is to write the quadratic equation in standard form by altering the given equation \(-x^2 = 15\). This involves changing the equation by multiplying both sides by \(-1\), turning the equation into \(x^2 = -15\).
Here, algebraic manipulation aids in restructuring the equation to fit a conventional format, which is critical to recognizing and solving these problems efficiently. This skill is not only useful in academics but also helps in logical thinking and problem-solving in various real-life situations.

• Simplifying equations to their base form
• Combining like terms and coefficients
• Adjusting signs and operations

Mastery in algebraic manipulation lets you handle equations with confidence, ensuring clarity and precision in problem-solving.

Equation Transformation

Equation transformation involves reworking an equation to achieve a specific form or simplify it. In the process of solving \(-x^2 = 15\), we employed transformation techniques to rewrite it in the standard form \(x^2 - 15 = 0\). The main steps were transforming the equation's sign and relocating the constant term to consolidate the expression.
To transform an equation effectively, one has to be adept at:

• Transitioning negative to positive values
• Shifting terms across the equals sign keeping the balance of the equation
• Reversing operations, such as addition and subtraction or multiplication and division

These changes ensure that the equation reflects a standard form \(ax^2 + bx + c = 0\), which is simpler to interpret and solve. Such transformations are essential for analyzing and understanding higher mathematical concepts efficiently.