Problem 84 - Elementary Algebra

Question

Problem: Use the quadratic formula to solve \(x^{2}-3 x-7=0\). $$ \text { Incorrect Answer: } \begin{aligned} x &=\frac{-3 \pm \sqrt{(-3)^{2}-4(1)(7)}}{2(1)} \\ x &=\frac{3 \pm \sqrt{9-28}}{2} \\ x &=\frac{3 \pm \sqrt{-19}}{2} \\ x &=\frac{3 \pm \sqrt{19} i}{2} \end{aligned} $$

Step-by-Step Solution

Answer

x = \frac{3 \, \text{\textpm} \, \sqrt{37}}{2}

Step 1: Identify the Coefficients

In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients \(a, b,\) and \(c\). For the equation \(x^2 - 3x - 7 = 0\), \(a = 1\), \(b = -3\), and \(c = -7\).

Step 2: Quadratic Formula

Recall the quadratic formula: \[ x = \frac{-b \, \text{\textpm} \, \sqrt{b^2 - 4ac}}{2a}. \]

Step 3: Substitute the Coefficients into the Quadratic Formula

Substitute \(a = 1\), \(b = -3\), and \(c = -7\) into the quadratic formula: \[ x = \frac{-(-3) \, \text{\textpm} \, \sqrt{(-3)^2 - 4(1)(-7)}}{2(1)} \]

Step 4: Simplify Inside the Square Root

Simplify the expression inside the square root: \((-3)^2 - 4(1)(-7) = 9 + 28 = 37\). So, the equation becomes: \[ x = \frac{3 \, \text{\textpm} \, \sqrt{37}}{2} \]

Step 5: Final Solution

Thus, the solutions of the quadratic equation \(x^2 - 3x - 7 = 0\) are: \[ x = \frac{3 + \sqrt{37}}{2} \] and \[ x = \frac{3 - \sqrt{37}}{2} \]

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). These equations can have real or complex solutions, often referred to as 'roots' or 'solutions' of the equation. One of the most efficient methods to solve quadratic equations is by using the quadratic formula. This formula is a straightforward way to find the solutions without needing to factorize the polynomial or complete the square.
To use the quadratic formula, you must first identify the coefficients in your quadratic equation. In the given problem \(x^2 - 3x - 7 = 0\), the coefficients are: \(a = 1\), \(b = -3\), and \(c = -7\).
Substitute these values into the quadratic formula: \[ x = \frac{-b \, \text{\textpm} \, \sqrt{b^2 - 4ac}}{2a} \] and simplify the resulting expression.

Coefficients

In any polynomial equation, coefficients are the numerical factors that scale the variables. For quadratic equations of the form \(ax^2 + bx + c = 0\), the coefficients are \(a\), \(b\), and \(c\).
It is important to correctly identify these coefficients before substituting them into the quadratic formula. In our example \(x^2 - 3x - 7 = 0\), the coefficients are straightforward:

• \(a = 1\)
• \(b = -3\)
• \(c = -7\)

The correct substitution ensures accurate calculation of the roots.
An error in identifying coefficients can lead to incorrect solutions, as seen in the step-by-step process provided earlier.

Complex Numbers

Complex numbers are numbers that have both real and imaginary parts. A complex number is generally represented in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
In our earlier incorrect solution, we encountered the term \(\sqrt{-19} = \sqrt{19}i\). This term is a complex number because the negative sign under the square root results in an imaginary number.
However, in our correct solution with \[ x = \frac{3 \, \text{\textpm} \, \sqrt{37}}{2} \], the value under the square root is positive, leading to real roots.

Simplification

Simplification is a crucial step in solving quadratic equations using the quadratic formula. It involves breaking down expressions into simpler forms to make calculations easier and more accurate.
For example, in the solution of \(x^2 - 3x - 7 = 0\), the expression inside the square root is simplified as follows:

• First, compute \(b^2 = (-3)^2 = 9\)
• Then, compute \(-4ac = -4(1)(-7) = 28\)
• Add these together to get \(9 + 28 = 37\)

Thus, the simplified expression inside the square root is \(\sqrt{37}\), leading to the final solutions:
\[ x = \frac{3 + \sqrt{37}}{2} \] and \[ x = \frac{3 - \sqrt{37}}{2} \].

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) for which the equation \(ax^2 + bx + c = 0\) holds true. There can be two, one, or no real roots depending on the discriminant \(b^2 - 4ac\).
For \(x^2 - 3x - 7 = 0\), calculating the discriminant gives \(b^2 - 4ac = 37\). Since this is positive, the quadratic equation has two distinct real roots.
Using the quadratic formula, you find these roots to be:

• \[ x = \frac{3 + \sqrt{37}}{2} \]
• and \[ x = \frac{3 - \sqrt{37}}{2} \]

These solutions provide the values of \(x\) where the original equation \(x^2 - 3x - 7 = 0\) becomes true.