Problem 91 - Algebra 2

Question

Solve using the Quadratic Formula. \(x^{2}-12 x+25=0\)

Step-by-Step Solution

Answer

The solutions of the equation \(x^{2}-12 x+25=0\) are \(x1 = 5 + 2\sqrt{11}\) and \(x2 = 5 - 2\sqrt{11}\).

Step 1: Identify \(a\), \(b\), and \(c\)

In our equation, \(x^{2}-12 x+25=0\), we can see that \(a = 1\), \(b = -12\), and \(c = 25\).

Step 2: Substitute \(a\), \(b\), and \(c\) into the quadratic formula

Now we substitute in the values for our equation into the quadratic formula which gives us: \(x = \frac{-(-12) \pm \sqrt{(-12)^{2}-4(1)(25)}}{2(1)}\).

Step 3: Calculate the value under the square root

The value under the square root, also known as the discriminant, can be evaluated as: \((-12)^{2}-4(1)(25) = 144-100 =44\). So the equation becomes: \(x = \frac{12 \pm \sqrt{44}}{2}\).

Step 4: Simplify the equation to find the solutions

Finally, solving the equation, \(x = \frac{12 \pm \sqrt{44}}{2}\) yields \(x1 = 5 + 2\sqrt{11}\) and \(x2 = 5 - 2\sqrt{11}\).

Key Concepts

Understanding the Discriminant

The discriminant is a crucial part of solving quadratic equations by using the quadratic formula. It is the value found under the square root in the formula: \[ b^2 - 4ac \] The discriminant serves as an indicator of the types of solutions a quadratic equation might have.

• If the discriminant is positive, you'll have two distinct real roots.
• If the discriminant is zero, there will be exactly one real root, meaning the parabola touches the x-axis at a single point.
• If it is negative, no real roots exist, and the roots are complex numbers.

In our exercise, the discriminant is \( 44 \), which is positive. This means the equation \( x^2 - 12x + 25 = 0 \) indeed has two distinct real roots that students can find using further calculations.

Demystifying Quadratic Equations

A quadratic equation is any equation that takes the form: \[ ax^2 + bx + c = 0 \] where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). This ensures the equation involves a squared term, making it quadratic.
The roots (solutions) of a quadratic equation are the values of \( x \) that make the equation true. These roots can be found using various methods like factoring, completing the square, or using the quadratic formula. The method varies based on the equation and the values it holds for \( a \), \( b \), and \( c \).
The equation \( x^2 - 12x + 25 = 0 \) is a perfect example of a quadratic equation where parameters have a distinct role, with \( a = 1, b = -12, \) and \( c = 25 \). Identifying these correctly is the first step towards finding the equation's roots efficiently.

Step-by-Step Solution Clarified

Solving a quadratic equation using the quadratic formula requires a clear step-by-step process. Here, we illustrate the methodical approach:

• **Step 1**: Identify the coefficients \( a, b, \), and \( c \). In our equation, these are \( a = 1 \), \( b = -12 \), and \( c = 25 \).
• **Step 2**: Plug these values into the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For this equation, it became: \[ x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \times 1 \times 25}}{2 \times 1} \]
• **Step 3**: Solve the discriminant \( b^2 - 4ac \). We calculated \( 44 \).
• **Step 4**: Simplify the equation \( x = \frac{12 \pm \sqrt{44}}{2} \), which gave us the solutions \( x_1 = 5 + 2\sqrt{11} \) and \( x_2 = 5 - 2\sqrt{11} \).

Breaking down the process helps assurances students to understand each calculation step rather than just applying the formula mechanically.