Problem 76 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3 + \sqrt{37}}{2}\) and \(x = \frac{3 - \sqrt{37}}{2}\).

Step 1: Identify coefficients

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

Step 2: Write the quadratic formula

The quadratic formula to solve \(ax^2 + bx + c = 0\) is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Substitute the coefficients

Substitute the values \(a = 1, b = -3, c = -7\) into the quadratic formula: \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-7)}}{2(1)}\).

Step 4: Simplify the equation

Simplify the expression under the square root (the discriminant): \(x = \frac{3 \pm \sqrt{9 + 28}}{2}\).

Step 5: Calculate the discriminant

Compute the value inside the square root: \(x = \frac{3 \pm \sqrt{37}}{2}\).

Step 6: Find the solutions

Simplify the solutions: \(x = \frac{3 + \sqrt{37}}{2}\) and \(x = \frac{3 - \sqrt{37}}{2}\).

Key Concepts

Solving Quadratic Equations

To solve a quadratic equation like the one in the exercise, we use the quadratic formula. A quadratic equation typically looks like this: \(ax^2 + bx + c = 0\), where:

• \texttt{a} is the coefficient of \(x^2\) (the quadratic term)
• \texttt{b} is the coefficient of \(x\) (the linear term)
• \texttt{c} is the constant term (no \(x\))

The quadratic formula is one of the most efficient methods to find the solutions to these types of equations. In our specific problem \(x^2 - 3x - 7 = 0\), we need to find the values of \(x\) that satisfy this equation.

Coefficients in Quadratic Equations

Understanding the coefficients \(a, b, c\) in the equation is crucial for using the quadratic formula accurately. Look at the equation \(x^2 - 3x - 7 = 0\):

• \texttt{a} is the coefficient of \(x^2\), which is \(1\).
• \texttt{b} is the coefficient of \(x\), and it is \(-3\).
• \texttt{c} is the constant term, which is \(-7\).

Knowing these coefficients ensures we use the correct numbers when substituting into the quadratic formula.

Discriminant in Quadratic Equations

The discriminant is a key part of the quadratic formula and helps determine the nature of the equation's roots. The discriminant is inside the square root of the quadratic formula: \(\frac{-b \textpm \textsqrt{b^2 - 4ac}}{2a}\) For the problem \(x^2 - 3x - 7 = 0\): \texttt{b = -3}, \texttt{a = 1}, \texttt{c = -7}First, compute the discriminant \(b^2 - 4ac\) → \((-3)^2 - 4 \times 1 \times (-7) = 9 + 28 = 37\). Since the discriminant is positive, the quadratic equation has two distinct real solutions.

Steps to Use Quadratic Formula

To effectively use the quadratic formula to solve the equation \(x^2 - 3x - 7 = 0\), follow these steps:

• Identify the coefficients: \texttt{a = 1}, \texttt{b = -3}, \texttt{c = -7}.
• Write the quadratic formula: \(x = \frac{-b \textpm \textsqrt{b^2 - 4ac}}{2a}\).
• Substitute the coefficients: \(x = \frac{-(-3) \textpm \textsqrt{(-3)^2 - 4 \times 1 \times (-7)}}{2 \times 1}\).
• Simplify the equation: \(x = \frac{3 \textpm \textsqrt{9 + 28}}{2}\)
• Calculate the discriminant: \(x = \frac{3 \textpm \textsqrt{37}}{2}\)
• Find the final solutions: \(x = \frac{3 + \textsqrt{37}}{2}\) \texttt{and} \(x = \frac{3 - \textsqrt{37}}{2}\)

Practicing these steps helps reinforce your understanding and application of the quadratic formula!