Problem 84 - Algebra 2

Question

Rewrite each equation in vertex form. $$ y=-2 x^{2}+x-10 $$

Step-by-Step Solution

Answer

The equation in vertex form is \(y = -2(x - 0.25)^2 - 9.875\).

Step 1: Factor out the Coefficient

First, we factor out -2 from the x terms. So, we rewrite the equation as \(y = -2(x^2 - 0.5x) - 10\).

Step 2: Complete the Square

In order to complete the square, we add and subtract \((0.5/2)^2 = 0.0625\) within the parentheses. Doing so, the equation becomes \(y = -2[(x^2 - 0.5x + 0.0625) - 0.0625] - 10\).

Step 3: Rewrite as a Perfect Square

The expression inside the square brackets is now a perfect square trinomial which can be rewritten as: \(y = -2[(x - 0.25)^2 - 0.0625] - 10\).

Step 4: Expand Outside the Parentheses

We now distribute the -2 to the terms inside the square brackets: \(y = -2(x - 0.25)^2 + 0.125 - 10\).

Step 5: Simplify to Obtain Vertex Form

Finally, we simplify the equation to its vertex form: \(y = -2(x - 0.25)^2 - 9.875\).

Key Concepts

Quadratic Equation

The quadratic equation is a type of polynomial equation that takes the general form: \(ax^2 + bx + c = 0\). It is called "quadratic" because the highest exponent of the variable \(x\) is 2, which makes it a degree 2 polynomial.

Some important characteristics include:

• The curve produced by a quadratic equation is known as a parabola.
• Parabolas can open upwards or downwards depending on the coefficient \(a\).
• The vertex form of a quadratic equation helps identify the vertex point—that point which is either the peak or the trough of the parabola.

Quadratic equations are foundational in algebra, and solving them helps in understanding parabolas, optimizing functions, and more. To solve or rewrite these equations efficiently, techniques like completing the square can be very useful.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that makes it easier to understand and solve. This transformation rewrites the quadratic in a way that makes the vertex of the parabola easily identifiable. The vertex form of a quadratic equation is \(y = a(x-h)^2 + k\), where \(h\) and \(k\) represent the coordinates of the vertex.

Here’s how completing the square works:

• First, factor out the leading coefficient if necessary, from the squared term, focusing on the terms involving \(x\).
• Next, take half of the coefficient of the \(x\) term, square it, add it within the expression, and balance it out by subtracting this same value.
• This process creates a perfect square trinomial inside the parenthesis.

Completing the square is a useful skill not just for rewriting equations in vertex form, but also for solving quadratic equations and understanding their properties.

Factoring

Factoring is a mathematical process used to express a polynomial as a product of its factors. It simplifies polynomials and allows us to find solutions to quadratic equations. The method of factoring depends on identifying pairs of numbers whose product and sum relate to the coefficients in the polynomial equation.

In the context of quadratics, here’s a simple guide:

• Express the quadratic equation in the form \(ax^2 + bx + c\).
• Identify the greatest common factor if present and factor it out.
• Determine two numbers that multiply to give \(ac\) and sum to give \(b\). These numbers are crucial for factoring expressions.
• Rewrite the equation using these numbers to decompose the middle term, which can then be paired and factored further.

It’s important to master factoring, as it aids in quickly solving equations and simplifying expressions, especially when dealing with quadratic equations.

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression that's the result of squaring a binomial. It can be easily rewritten using the expression \((a+b)^2 = a^2 + 2ab + b^2\) or \((a-b)^2 = a^2 - 2ab + b^2\).

Characteristics of perfect square trinomials include:

• They consist of three terms shaped by squaring a sum or a difference.
• The first and last terms are perfect squares themselves.
• The middle term is twice the product of the terms in the binomial.

These expressions are very advantageous when completing the square in a quadratic equation. Recognizing and rewriting expressions as perfect square trinomials simplifies solving equations and graphing parabolas by making the structure of the equation clearer.