Problem 28 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ x^{2}-9 x=-14 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 7\) and \(x = 2\)

Step 1: Rearrange the equation

The given equation is \(x^{2}-9x = -14\). First, re-arrange the equation into standard form, which is \(ax^2 + bx + c = 0\). So, by adding 14 on both sides, the equation becomes \(x^{2}-9x +14 = 0\).

Step 2: Factoring the quadratic equation

The quadratic equation can be factored by finding two numbers that multiply to 14 (the 'c' term) and add to -9 (the coefficient of 'x'). These two numbers are -7 and -2, because -7*-2=14 and -7-2=-9. So, the factored equation is \((x-7)(x-2) = 0\).

Step 3: Solving for x

To solve for 'x', we set each factor equal to zero and solve. This gives us \(x-7 = 0\) and \(x-2 = 0\), and solving these gives \(x = 7\) and \(x = 2\) as the solutions to the equation.

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations. It involves expressing a quadratic expression as a product of simpler binomials. For instance, when solving the quadratic equation \(x^2 - 9x + 14 = 0\), the goal is to rewrite it as a product of two binomials that set to zero when equal. The idea is to find two numbers, known as factors, that can multiply to the constant term 'c' (in this case, 14) but also add up to the coefficient of 'x' (which is -9).

• Identify the 'a', 'b', and 'c' in the quadratic expression \(ax^2 + bx + c\).
• Find two numbers that multiply to 'c' and add to 'b'.
• Rewrite the quadratic as a product of two binomials.

By finding these numbers, -7 and -2, the quadratic \(x^2 - 9x + 14\) can be factored as \((x-7)(x-2)\). Solving this involves setting each factor equal to zero.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is crucial for understanding how to approach solving it. A quadratic equation in this form is expressed as \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants, and 'x' represents the variable or unknown we wish to solve for. Understanding this form makes it straightforward to apply methods such as factoring, the quadratic formula, or completing the square to find the solutions or roots. By rearranging any given quadratic equation into this standardized format, you can easily identify the values needed for any solving method. Consider the original equation \(x^2 - 9x = -14\), here you would:

• Move all terms to one side of the equation to form \(x^2 - 9x + 14 = 0\).
• Ensure the equation maintains the structure of \(ax^2 + bx + c = 0\).

Once in standard form, you can proceed with various solving techniques, such as factoring.

Roots of a Quadratic Equation

The roots of a quadratic equation are the solutions for 'x' that make the equation true or zero. These roots can often be found by factoring the quadratic expression and solving for when each factor equals zero. Given a factored quadratic equation such as \((x-7)(x-2) = 0\), the roots are the values of 'x' that solve \(x-7 = 0\) and \(x-2 = 0\). Solving these gives:

• For \(x-7 = 0\), add 7 to both sides to get \(x = 7\).
• For \(x-2 = 0\), add 2 to both sides to get \(x = 2\).

These values, \(x = 7\) and \(x = 2\), are called the roots of the quadratic equation \(x^2 - 9x + 14 = 0\). These roots are the points where the parabola, represented by the quadratic equation, intersects the x-axis.