Problem 50 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}+14 x-7=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}+14x-7=0\) are \(x = 0.27\) and \(x = -26.27\)

Step 1 Identification

Identify the values of \(a\), \(b\), and \(c\) in our quadratic equation. In this equation, \(x^{2}+14x-7=0\), \(a = 1\), \(b = 14\), and \(c = -7\).

Step 2 Quadratic Formula

Apply the values of \(a\), \(b\), and \(c\) to the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substituting the values, we get: \(x = \frac{-14 \pm \sqrt{14^2 - 4*1*-7}}{2*1}\).

Step 3 Simplify

Simplify the expression under the square root first: \(14^2 - 4*1*-7 = 196 + 28 = 224\). Then substitute back into the equation, to get two potential solutions for \(x\): \(x = \frac{-14 \pm \sqrt{224}}{2}\). Further simplifying this, we get the solutions \(x = 0.27\) and \(x = -26.27\)

Key Concepts

quadratic formula

Let's delve into the quadratic formula, a powerful tool for solving quadratic equations. A quadratic equation is any equation that can be put in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our example, the equation is \(x^2 + 14x - 7 = 0\). This implies that \(a = 1\), \(b = 14\), and \(c = -7\).

The quadratic formula is used to find the values of \(x\) that satisfy the equation. It is expressed as:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula allows us to compute the roots of any quadratic equation directly and efficiently. Understanding and applying this formula is crucial, as it provides an exact solution for \(x\) without trial and error.

Use the quadratic formula whenever the quadratic equation cannot easily be factored, or to verify solutions obtained by other means.

solving equations

When solving equations, especially quadratics like \(x^2 + 14x - 7 = 0\), it's important to follow a systematic approach. Let's outline the steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation. For our equation: \(a = 1\), \(b = 14\), and \(c = -7\).
• Substitute these values into the quadratic formula:

\[x = \frac{-14 \pm \sqrt{14^2 - 4 \cdot 1 \cdot -7 }}{2 \cdot 1}\]

• Simplify the expression under the square root, otherwise known as the discriminant. Here, \(14^2 - 4 \times 1 \times -7 = 196 + 28 = 224\).
• This results in two potential solutions for \(x\):

\[x = \frac{-14 \pm \sqrt{224}}{2}\] The calculation for the discriminant is crucial, as it determines the number and type of solutions. A positive discriminant indicates two real and distinct solutions. This systematic problem-solving approach is applicable to all types of quadratic equations.

mathematical solutions

Mathematical solutions to quadratic equations involve careful computation and interpretation. In our example, the end result of substituting and simplifying the quadratic formula gives us two potential solutions for \(x\) as:
\[x = \frac{-14 \pm \sqrt{224}}{2}\]
Simplifying further, \(\sqrt{224} \approx 14.97\). Thus, the solutions become:

• \(x = \frac{-14 + 14.97}{2} \approx 0.27\)
• \(x = \frac{-14 - 14.97}{2} \approx -26.27\)

These solutions indicate where the quadratic equation intersects the x-axis on a graph, representing the roots of the equation. Each solution is valuable and offers insight into real-world applications of quadratic equations.

Always double-check your calculations to ensure accuracy, especially when it involves square roots or division. By confirming each step, you cement your understanding and build problem-solving confidence.