Problem 10 - Elementary Algebra

Question

Solve each quadratic equation by factoring. $$ 2 x^{2}=-2-5 x $$

Step-by-Step Solution

Answer

Answer: The solutions for the quadratic equation $$2x^2 = -2 - 5x$$ are $$x = -2$$ and $$x = -\frac{1}{2}$$.

Step 1: Rewrite the equation in standard form

Move the terms from the right side of the equation to the left side to write the equation in the standard form $$ax^2 + bx + c = 0$$: $$ 2x^2 + 5x + 2 = 0 $$

Step 2: Factor the equation

Now, we will factor the equation. Try to find two binomials whose product equals the given quadratic equation. In this case, the binomials are $$(x+2)$$ and $$(2x+1)$$: $$ (2x^2 + 5x + 2) = (x+2)(2x+1) $$

Step 3: Solve for x

To solve for x, set each binomial equal to zero: $$ x + 2 = 0 \quad \mathrm{or} \quad 2x + 1 = 0 $$

Step 4: Find the solutions

Solve each equation from Step 3 individually: For $$x + 2 = 0$$, we get $$x = -2$$. For $$2x + 1 = 0$$, we get $$x = -\frac{1}{2}$$. Thus, the solutions for the quadratic equation $$2x^2 + 5x + 2 = 0$$ are $$x = -2$$ and $$x = -\frac{1}{2}$$.

Key Concepts

Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable x, with a non-zero coefficient for the x² term. It has the general form of (ax^2 + bx + c = 0), where 'a' is not equal to zero. Here, 'a', 'b', and 'c' are constants, with 'a' being the coefficient of the squared term, 'b' the coefficient of the linear term, and 'c' the constant term. Solving a quadratic equation involves finding the values of x that make the equation true, and these values are known as the roots of the equation.
Quadratic equations can be solved by various methods, including factoring, completing the square, using the quadratic formula, or graphing. Factoring is one of the simplest methods and helps to express the quadratic equation as a product of two binomials. Each method has its own advantages, but factoring is often the quickest and most intuitive, especially when the solutions are rational numbers.

Factoring Binomials

Factoring is the process of breaking down a composite number or a polynomial into a product of smaller numbers or polynomials that, when multiplied together, will give the original number or polynomial. In the context of solving quadratic equations, factoring binomials refers to finding two binomial expressions that, when multiplied together, result in the original quadratic polynomial. This is often done by looking for a pair of numbers that add up to 'b' (the coefficient of the x term) and multiply to 'ac' (the product of 'a' and 'c').
The key to successful factoring is identifying patterns or using techniques such as the 'ac method' or 'trial and error'. Factoring is particularly effective when the roots of the quadratic equation are rational numbers, allowing for a more straightforward and cleaner solution. The ability to factor swiftly can vastly simplify solving quadratic equations and is a fundamental skill in algebra.

Standard Form Quadratic

The standard form of a quadratic equation is written as (ax^2 + bx + c = 0). This form provides a clear blueprint for identifying the coefficients needed for certain solution methods, such as factoring or applying the quadratic formula. It specifies that the x² term is followed by an x term and then a constant.
When working with quadratic equations, it is often necessary to rearrange terms to achieve this standard form before proceeding with factoring or other solution methods. One common step in this process is moving all terms to one side of the equation, setting it equal to zero. This manipulation is essential as it sets the stage for factoring and for the use of other solution strategies. Understanding the standard form is crucial as it lays the groundwork for different approaches to find the solutions of the quadratic equation.