Lesson 2: Some Quadratic Models

New Concept

We'll learn a powerful new method to solve quadratic equations. By rewriting the equation to equal zero and then factoring, we can use the Zero-Factor Principle to find the solutions. This breaks a tough problem into simple steps.

What’s next

Get ready to see this in action. We'll start with real-world revenue models, then master factoring and solving step-by-step with interactive practice cards.

Property

Revenue is the amount of money a company takes in from selling a product. To find the total revenue from the sale of a product, we multiply the price of one item by the number of items sold.

Revenue = (price per item) · (number of items sold)

Examples

• A bake sale sells cookies for $p$ dollars each. They find that they sell $50 - 2p$ cookies. The revenue is $R = p(50 - 2p) = 50p - 2p^2$.

Property

If the product of two numbers is zero, then one (or both) of the numbers must be zero. Using symbols,

If $AB = 0$, then either $A = 0$ or $B = 0$.

Examples

• To solve $(x - 7)(x + 3) = 0$, set each factor to zero. $x - 7 = 0$ gives $x = 7$, and $x + 3 = 0$ gives $x = -3$.

Property

Factoring is the reverse of multiplying factors together. To factor an expression with a common factor, find the largest factor that divides into each term and use the distributive law in reverse. A common factor is an expression that is a factor of each term of another expression.

Examples

• To factor $10x^2 - 15x$, the largest common factor is $5x$. Dividing each term by $5x$ gives $2x$ and $-3$. The factored form is $5x(2x - 3)$.

• In the expression $18y^2 + 24y$, the largest common factor is $6y$. Factoring this out gives $6y(3y + 4)$.

Property

To Solve a Quadratic Equation by Factoring:

1. Write the equation with zero isolated on the right side.

2. Factor the left side of the equation.

Property

We should never divide both sides of an equation by the variable, because we risk losing one of the solutions. This mistake happens when the variable you are dividing by could be equal to zero.

Examples

• To solve $x^2 = 7x$, dividing by $x$ incorrectly gives $x=7$. The correct method is $x^2 - 7x = 0$, so $x(x-7)=0$, which gives both solutions: $x=0$ and $x=7$.

• In the equation $4a^2 = -8a$, if you divide by $4a$, you get $a = -2$ and lose the $a=0$ solution. Factoring gives $4a(a+2)=0$, so $a=0$ or $a=-2$.