Problem 58 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation using any method. Find only real solutions. $$x^{2}-4 x=-4$$

Step-by-Step Solution

Answer

The real solution for the quadratic equation \(x^{2}-4x = -4\) is \(x=2\).

Step 1: Rearrange the equation in standard form

At first rearrange the equation to standard form of quadratic equation. Here, that means moving \( -4 \) from the right side of the equation to the left: \(x^{2} - 4x + 4 = 0\).

Step 2: Apply the quadratic formula

Afterwards apply the quadratic formula \(x = \frac{-b ± \sqrt{b^{2}-4ac}}{2a}\). Here, \(a=1\), \(b=-4\), and \(c=4\). Apply it to find the values of \(x\).

Step 3: Compute the result

Substitute, \(a=1\), \(b=-4\), \(c=4\) into the quadratic formula to get \(x = \frac{-(-4) ± \sqrt{(-4)^{2}-4*1*4}}{2*1} = \frac{4 ± \sqrt{0}}{2}\). As the square root of 0 is 0, it results only into one root instead of two (\(x= \frac{4±0}{2} = 2\))

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations typically in the form \(ax^2 + bx + c = 0\). This is especially helpful when factoring the equation is difficult or impossible. The formula is given by:

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

In this equation:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
• \(x\) represents the solutions to the equation.
• The "\(\pm\)" symbol indicates that there can be two solutions, one with addition and one with subtraction of the square root part.

To use the quadratic formula, substitute the values of \(a\), \(b\), and \(c\) into the formula. This will result in the values of \(x\), revealing the solution(s) to the equation. It's useful to first ensure the equation is in standard form, as this clearly identifies \(a\), \(b\), and \(c\).

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is crucial as it sets the stage for using the quadratic formula or other methods effectively. The standard form is:

• \(ax^2 + bx + c = 0\)

Here’s why it's important:

• It organizes the equation so you can easily identify the coefficients \(a\), \(b\), and \(c\).
• It must equal zero, because this ensures all terms are aligned correctly for solving.

To convert any quadratic equation to standard form, ensure that all terms are on one side of the equation.
In the example provided, converting \(x^2 - 4x = -4\) to standard form involves moving \(-4\) to the left to get \(x^2 - 4x + 4 = 0\). This puts the equation in a solvable format, making it easier to apply solution methods like the quadratic formula.

Real Solutions

When solving quadratic equations, one critical aspect is determining the type of solutions, particularly real solutions. Real solutions are values of \(x\) that satisfy the quadratic equation and can be graphically represented on a number line.
The nature of the solutions largely depends on the discriminant, calculated as \(b^2 - 4ac\), from the quadratic formula. Here’s what different discriminant results mean:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is one real solution. This occurs because the square root part becomes zero, resulting in a single value for \(x\).
• If \(b^2 - 4ac < 0\), there are no real solutions, only complex ones.

In our example, using the quadratic formula on \(x^2 - 4x + 4 = 0\) results in a discriminant of zero (\(\sqrt{0}\)), so there is exactly one real solution: \(x = 2\). Understanding whether solutions are real is key to knowing how the equation behaves and what kind of answers to expect.