Problem 66 - Algebra A Combined Function

Question

Solve for \(x\). $$ x^{2}+4 x=12 $$

Step-by-Step Solution

Answer

The solutions are \( x = 2 \) and \( x = -6 \).

Step 1: Rearrange the Equation

To solve the quadratic equation, start by rearranging all terms to one side so that we have: \[ x^2 + 4x - 12 = 0 \] This is the standard form of a quadratic equation, \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 4 \), and \( c = -12 \).

Step 2: Use the Quadratic Formula

The quadratic formula is \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Insert \( a = 1 \), \( b = 4 \), and \( c = -12 \) into the formula:\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \].

Step 3: Simplify Under the Square Root

Calculate the discriminant, \( b^2 - 4ac \):\[ 4^2 - 4 \cdot 1 \cdot (-12) = 16 + 48 = 64 \].

Step 4: Complete the Square Root Calculation

Simplify further using the square root of the discriminant:\[ \sqrt{64} = 8 \]. So the equation becomes: \[ x = \frac{-4 \pm 8}{2} \].

Step 5: Solve for Both Possible Values of x

Calculate the two possible values of \( x \):1. \( x = \frac{-4 + 8}{2} = \frac{4}{2} = 2 \)2. \( x = \frac{-4 - 8}{2} = \frac{-12}{2} = -6 \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \). The formula itself is:

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

This formula allows you to find the values of \( x \) that make the equation true. Here, \( a \), \( b \), and \( c \) are coefficients in the equation, specifically:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

The quadratic formula includes the "±" symbol, giving us two potential solutions for \( x \): one with addition and the other with subtraction under the square root. Try using the formula whenever you see a quadratic equation to find its roots!
Always remember to substitute \( a \), \( b \), and \( c \) directly into the formula to solve for \( x \).

Discriminant

The discriminant is a significant part of the quadratic formula. It appears under the square root and is calculated as \( b^2 - 4ac \).
The value of the discriminant provides us crucial information about the nature of the roots of the quadratic equation:

• If the discriminant is positive, \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), the quadratic equation has exactly one real root, or a repeated root.
• If the discriminant is negative, \( b^2 - 4ac < 0 \), then the quadratic equation has no real roots, but two complex roots.

In our specific example, the discriminant was calculated as 64, a positive number, implying the equation has two real and distinct roots. Understanding the discriminant can greatly simplify the process of solving quadratic equations by predicting the nature of the solutions.

Standard Form of Quadratic Equation

Getting an equation into the standard form of a quadratic equation is the first crucial step in solving using the quadratic formula. This form is expressed as \( ax^2 + bx + c = 0 \) and sets the stage for all following calculations.

• \( a \) is the coefficient in front of \( x^2 \)
• \( b \) is the coefficient in front of \( x \)
• \( c \) is the constant term

To convert any quadratic equation into standard form, rearrange all terms onto one side of the equation. For example, an equation like \( x^2 + 4x = 12 \) needs to be rearranged as \( x^2 + 4x - 12 = 0 \) by moving the 12 to the left side. Now, you have the necessary form to identify the coefficients \( a \), \( b \), and \( c \) and proceed with the quadratic formula or other solution methods. Knowing how to express an equation in its standard form is essential for understanding how quadratic equations are structured and solved.