Problem 60 - Elementary Algebra

Question

\(x^{2}-3 x+5=0\)

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3 \pm i \sqrt{11}}{2} \).

Step 1: Identify the quadratic equation

The given equation is a quadratic equation in the form of \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -3\), and \(c = 5\).

Step 2: Apply the quadratic formula

To solve it, use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].Substitute \(a = 1\), \(b = -3\), and \(c = 5\) into the formula.

Step 3: Simplify under the square root

Calculate the discriminant, \(b^2 - 4ac\):\[ (-3)^2 - 4(1)(5) = 9 - 20 = -11 \].

Step 4: Determine the nature of the roots

Since the discriminant is negative (\(-11\)), the roots are complex. The expression under the square root will be simplified to imaginary numbers.

Step 5: Finalize the solution

Using the quadratic formula with a negative discriminant:\[ x = \frac{3 \pm \sqrt{-11}}{2(1)} \]This simplifies to:\[ x = \frac{3 \pm i\sqrt{11}}{2} \].

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides the solutions (also known as roots) to any quadratic equation. Here's what each symbol represents:

• a: The coefficient of \(x^2\)
• b: The coefficient of \(x\)
• c: The constant term

Here's how you can use it:

• Identify \(a\), \(b\), and \(c\) from your quadratic equation.
• Substitute these values into the quadratic formula.
• Simplify the expression to find the values of \(x\).

For example, with the equation \(x^2 - 3x + 5 = 0\), we have \(a = 1\), \(b = -3\), and \(c = 5\). Plugging these into the formula, we get:\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(5)}}{2(1)} \]Which simplifies to:\[ x = \frac{3 \pm \sqrt{-11}}{2} \]

Complex Roots

In some cases, the quadratic equation does not have real number solutions. This usually happens when the discriminant (the part of the quadratic formula under the square root, \(b^2 - 4ac\)) is negative. A negative discriminant results in complex roots.Complex roots involve the imaginary unit \(i\), where \(i\) is defined as \(\sqrt{-1}\). When the discriminant is negative, we simplify the roots by factoring out \(i\). For the example \(x^2 - 3x + 5 = 0\), we found the discriminant to be \(-11\), which is negative. Thus, our solutions are complex:\[ x = \frac{3 \pm \sqrt{-11}}{2} \]This further simplifies to:\[ x = \frac{3 \pm i\sqrt{11}}{2} \]So, the two complex roots are \(\frac{3 + i\sqrt{11}}{2}\) and \(\frac{3 - i\sqrt{11}}{2}\).

Discriminant

The discriminant is an important part of the quadratic formula, given by \(b^2 - 4ac\). It helps determine the nature of the roots of a quadratic equation. Here's a quick guide to understanding the discriminant's significance:

• Positive Discriminant: If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• Zero Discriminant: If \(b^2 - 4ac = 0\), the quadratic equation has exactly one real root (or a repeated root).
• Negative Discriminant: If \(b^2 - 4ac < 0\), the quadratic equation has two complex roots.

For the equation we are discussing, \(x^2 - 3x + 5 = 0\), let's calculate the discriminant:\[ (-3)^2 - 4(1)(5) = 9 - 20 = -11 \]Since the discriminant is \(-11\), which is negative, the quadratic equation \(x^2 - 3x + 5 = 0\) has complex roots. Complex roots always come in conjugate pairs, meaning one will be the complex conjugate of the other.