Understanding Quadratic Equations: Factoring, Formula, and Applications

Study Notes

Understanding Quadratic Equations

A quadratic equation refers to a mathematical expression containing terms up to the second degree. These equations can be solved through various methods, including factoring, completing the square, and using the quadratic formula. In this article, we will discuss factoring, the quadratic formula, and solving quadratic equations through examples and real-world applications.

Factoring Quadratic Equations

Factoring is one method used to solve quadratic equations. This involves expressing the polynomial equation in terms of factors whose products are the coefficients of the original equation. An example of a quadratic equation that can be factored is:

(x - 2)^2 - (x - 1)^2 = 1

Expanding both sides of the equation, we obtain:

x^2 - 4x - x + 2 = x^2 - 2x + 1

Combining like terms, we get:

x^2 - 3x + 1 = 0

This is a perfect square trinomial, which can be factored into:

(x - 1)^2 = 0

The solutions for this equation would be (x = 1).

Quadratic Formula

The quadratic formula provides another method for solving quadratic equations. Given the general quadratic equation:

ax^2 + bx + c = 0

where (a), (b), and (c) are constants, the quadratic formula states:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For example, if we have the equation:

x^2 + 6x + 8 = 0

we can plug in (a=1), (b=6), and (c=8):

x = \frac{-6 \pm \sqrt{6^2 - 4(1)(8)}}{2(1)}
x = \frac{-6 \pm \sqrt{36 - 32}}{2}
x = \frac{-6 \pm \sqrt{4}}{2}
x = \frac{-6}{2} \text{ or } x = -\frac{\sqrt{4}}{2} = -\frac{1}{2}

Solving Word Problems with Quadratic Equations

Word problems involving quadratic equations typically require setting up an equation based on given information, then solving it using factoring or the quadratic formula. For example, consider the problem of finding the width of the matting around a wedding photo. The area of the resulting photo and matting is given as 1 square meter, which translates into 10,000 cm² (since 1cm = 0.01 m). The dimensions of the photo plus the matting are given as 80 cm + 2x and 60 cm + 2x. Setting up the equation for the area yields:

(L+2x)(W+2x)=1\text{ m}^2

Or, in cm:

(80\text{ cm }+2x)(60\text{ cm }+2x)=10,000\text{ cm}^2

Multiplying this out gives us:

4800+280x+4x^2=10,000

Which rearranges to:

4x^2+280x-5200=0

This equation can then be factored or solved using the quadratic formula to find the values of (x).

Description

Explore the concepts of factoring quadratic equations, using the quadratic formula, and applying these methods to solve real-world problems. Learn how to factor perfect square trinomials, apply the quadratic formula, and tackle word problems involving quadratic equations.