Problem 47 - Algebra 1: Concepts and Skills

Question

Solve \(x^{2}-9 x=36\) by factoring f. 12 and \(-3\) g. 12 and \(-3\) h. 4 and \(-9\) j. 9 and \(-4\)

Step-by-Step Solution

Answer

The solutions for the equation are 12 and -3. So, the correct choice is f.

Step 1: Rearrange the Equation

Rearrange the given equation \(x^{2}-9x = 36\) to match the quadratic equation format \(ax^{2} + bx + c = 0\). This is done by subtracting 36 on both sides, leading to \(x^{2}-9x-36=0\).

Step 2: Factorize the Quadratic Equation

Factor the quadratic equation into two binomial equations. \(x^{2}-9x-36=0\) can be factorized into \((x-12)(x+3)=0\).

Step 3: Solve for x

Setting each factor equal to zero leads to the solutions for x: \(x-12=0\) gets \(x=12\) and \(x+3=0\) gets \(x=-3\).

Key Concepts

Factoring Quadratic Equations

Factoring is a powerful algebraic tool used to solve quadratic equations, which are polynomial equations of the second degree, commonly in the form of ax^2 + bx + c = 0. The process of factoring involves breaking down the quadratic expression into a product of two binomials. For instance, to solve the equation x^2 - 9x = 36, you first rearrange it as x^2 - 9x - 36 = 0.

Then, you look for two numbers that multiply to give the product of the coefficient of x^2 (a) and the constant term (c), and also add up to the coefficient of x (b). These numbers are essential to factoring the equation as such: (x + m)(x + n) = 0, where m and n are the numbers you've found. In our example, 12 and -3 fit the bill, since 12 * -3 = -36 and 12 + (-3) = 9, which is the coefficient (with a negative sign) of x in our rearranged equation. Thus, the equation factorizes to (x - 12)(x + 3) = 0.

After successfully factoring, the product of the binomials is set to zero, and you solve for x individually from each binomial equation, leading to the solution x = 12 or x = -3.

Quadratic Equation Format

The standard format for any quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The second-degree term ax^2 indicates it is a quadratic and not a linear equation. Understanding this format is paramount in factoring and solving quadratic equations.

When approaching a quadratic equation like x^2 - 9x = 36, you must first manipulate the equation to fit this standard. By subtracting 36 from both sides, you receive x^2 - 9x - 36 = 0, which is now correctly formatted for factoring. This allows you to systematically address the equation, using known methods such as factoring, completing the square, or using the quadratic formula to find the values of x that satisfy the equation.

Binomial Equations

A binomial equation is an algebraic expression containing two distinct terms, such as (x + m) or (x - n). When solving by factoring, quadratic equations are broken down into two binomial equations, creating a situation where each binomial is set to zero in order to solve for x.

This process relies on the zero product property, where if the product of two factors is zero, at least one factor must be zero. Hence, for a factored form such as (x - 12)(x + 3) = 0, you derive the binomial equations x - 12 = 0 and x + 3 = 0. Solving these equations is straightforward: add 12 to both sides of the first to get x = 12, and subtract 3 from both sides of the second to get x = -3. These solutions are the roots of the original quadratic equation, giving you the values where the quadratic graph crosses the x-axis.