Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation using any convenient method. $$11 x^{2}+33 x=0$$

Step-by-Step Solution

Answer

The solutions for \(x\) in the equation \(11x^{2}+33x=0\) are \(x = 0\) and \(x = -3\).

Step 1: Factor out the greatest common factor

In this given equation \(11x^{2}+33x=0\), the greatest common factor (gcf) is \(11x\). When factored out, the equation becomes \(11x(x+3)=0\)

Step 2: Set each factor to zero and solve for x

Setting \(11x=0\), we find that one possible solution for x is 0. Setting \(x+3=0\), we find that the other possible solution for x is -3. Therefore, the equation \(11x^{2}+33x=0\) has two solutions.

Key Concepts

Factoring Quadratic Equations

Quadratic equations, which are polynomials of the form \(ax^{2}+bx+c=0\), are commonly solved by factoring, provided the equation can be factored. The process involves finding two binomials that when multiplied together, yield the original quadratic equation. To factor a quadratic equation, one must look for a pair of numbers that both add up to the coefficient \(b\) of the \(x\) term and multiply to give the product of the coefficient \(a\) and the constant term \(c\).

Factoring becomes simpler when a common factor is present in all terms of the quadratic equation. Factoring is akin to breaking down a complex problem into more manageable pieces—just as you would solve a puzzle by grouping similar pieces together before assembling the entire picture. When you factor the equation and apply the zero product property, you set the stage for efficiently finding the roots of the quadratic equation.

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the highest number that divides exactly into two or more numbers. In the context of quadratic equations, the GCF refers to the highest shared factor among the coefficients and constant term.

Identifying the GCF is a critical first step when factoring quadratic equations because it can simplify the equation, making the subsequent factoring steps more straightforward. By dividing each term by the GCF, the resulting expression is easier to deal with. Further, recognizing the GCF helps to avoid overlooking shared factors, which is essential for accurately factoring the equation and finding the correct solutions.

Zero Product Property

The zero product property is a fundamental principle in algebra that states if the product of two factors is zero, then at least one of the factors must be zero. This property is extremely useful when solving quadratic equations that have been factored into the form \( (a)(b) = 0 \).

After factoring the quadratic equation into two binomials, the zero product property allows us to set each factor equal to zero and solve for the variable. This leads to finding the roots or solutions of the equation. Stated simply, it transforms a multiplication problem into an easier addition or subtraction problem. This property is particularly valuable because it paves the way to directly solve for the unknown variable, making it a staple in the arsenal of methods for solving quadratic equations.