Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving a Quadratic Equation Find all real solutions of the equation. $$x^{2}-22 x+121=0$$

Step-by-Step Solution

Answer

The equation \(x^2-22x+121=0\) has two real roots, both are the same, \(x=11\).

Step 1: Identify a, b, and c from the quadratic equation

In the equation \(x^2-22x+121=0\), you can identify that \(a=1\) (coefficient of \(x^2\)), \(b=-22\) (coefficient of x), and \(c=121\). The equation is in the general form \(ax^2+bx+c=0\).

Step 2: Apply the quadratic formula

Plug the values of a, b, and c into the quadratic formula, \(x=(-b±\sqrt{b^2-4ac}) / 2a\). So, \(x=(22±\sqrt{(-22)^2-4(1)(121)}) / (2*1)\)

Step 3: Simplify the calculation under the square root in the formula

Compute the expression under the square root: \(b^2-4ac=(-22)^2-4*1*121=484-484=0\). Then the quadratic formula becomes \(x=(22±\sqrt{0}) / 2\).

Step 4: Compute the final values

Completing the calculation, \(x=(22±0) / 2\). This gives two solutions - identical in this particular case - which are \(x=22/2=11\).

Key Concepts

Quadratic Formula

The quadratic formula is a vital tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula helps find the solutions, or roots, of these equations. Here is how the quadratic formula is written: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. The formula provides two potential values for \(x\) because of the \(\pm\) symbol, thus accounting for the two possible solutions that a quadratic equation might have.
This formula is particularly useful when factoring is not possible, making it a preferred method for finding solutions. When using the quadratic formula, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\).
• Substitute these values into the quadratic formula.
• Simplify the expression to find the values of \(x\).

Understanding each part of the formula and how to navigate calculations will ensure finding the correct solutions.

Real Solutions

When solving any equation, finding real solutions refers to finding solutions that are real numbers—numbers without any imaginary component. With quadratic equations, solutions can be real or complex/imaginary, depending on the expression within the square root of the quadratic formula, known as the discriminant.
If an equation has:

• Real solutions, it means the roots are real numbers, either rational or irrational.
• Identical real solutions, it both confirms they're real and informs that both are the same.

In the provided example, the quadratic equation \(x^2 - 22x + 121 = 0\) has real solutions because the discriminant (the part under the square root in the quadratic formula) evaluates to zero, leading to identical roots. These roots are typically no different from each other, resulting in a single real solution when simplified.

Discriminant

The discriminant is the part of the quadratic formula located inside the square root, expressed as \(b^2 - 4ac\). Standard understanding of the discriminant helps in predicting the nature of a quadratic equation's solutions:

• If the discriminant is greater than zero, the equation has two distinct real solutions.
• If the discriminant is exactly zero, the equation has exactly one real solution, known as a repeated or double root.
• If the discriminant is less than zero, there are no real solutions; instead, solutions are complex (involving imaginary numbers).

In our given equation, \(x^2 - 22x + 121 = 0\), we found that \(b^2 - 4ac\) equals zero. This tells us the equation has a single real solution (specifically \(x = 11\)) since both potential solutions calculated through \( \pm \sqrt{0} \) converge to the same number. Understanding and calculating the discriminant is crucial for quickly determining the nature of roots in any quadratic equation.