Problem 12 - Algebra 2

Question

Solve each equation. State the number and type of roots. \(2 x^{2}-5 x+12=0\)

Step-by-Step Solution

Answer

The equation has two complex roots.

Step 1: Recognize the Equation Type

The given equation is a quadratic equation in the standard form \(ax^2 + bx + c = 0\) where \(a = 2\), \(b = -5\), and \(c = 12\).

Step 2: Calculate the Discriminant

The discriminant \(D\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute \(a = 2\), \(b = -5\), and \(c = 12\) into the formula: \[D = (-5)^2 - 4 \times 2 \times 12 = 25 - 96 = -71\].

Step 3: Analyze the Discriminant

Since the discriminant \(D = -71\) is less than zero, the quadratic equation \(2x^2 - 5x + 12 = 0\) has two complex roots. This means there are no real solutions.

Key Concepts

The Discriminant

When you're dealing with a quadratic equation, such as the one in our exercise, the concept of the discriminant is crucial. It's a tool that provides valuable information about the nature of the roots of the equation.

The discriminant is found using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients from the equation in its standard form \(ax^2 + bx + c = 0\). In the case of our exercise, \(a = 2\), \(b = -5\), and \(c = 12\).

Calculating the discriminant:

• Square of \(b\): \((-5)^2 = 25\)
• Product of \(4ac\): \(4 \times 2 \times 12 = 96\)
• Discriminant: \(25 - 96 = -71\)

A discriminant value helps determine the nature of the roots:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, or a repeated real root.
• If \(D < 0\), there are two complex roots, as shown in our exercise.

Complex Roots

Complex roots occur when a quadratic equation's discriminant is less than zero. In simpler terms, it means that the equation does not cross the x-axis and does not have any real number solutions.

Complex roots come in conjugate pairs, meaning they have the form:

• \(a + bi\)
• \(a - bi\)

where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, equating to \(\sqrt{-1}\). These roots are significant in mathematics because they allow us to work with negative square roots.

In the equation \(2x^2 - 5x + 12 = 0\), the negative discriminant tells us that the roots will be complex. Though we won't calculate them here, knowing the discriminant is negative is enough to understand the nature of the roots.

Quadratic Formula

The quadratic formula is a handy tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). It's particularly useful when factorization isn't feasible or if the roots are not simple integers.

The formula is defined as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to find the roots of the quadratic equation directly by substituting the values of \(a\), \(b\), and \(c\).

For our quadratic equation \(2x^2 - 5x + 12 = 0\), you can apply the quadratic formula to find the exact complex roots. Remember, because the discriminant is negative, the expression inside the square root becomes a complex number:

• The term \(\sqrt{b^2 - 4ac}\) turns into \(\sqrt{-71}\) in our scenario, which involves imaginary numbers.
• The overall result will include imaginary units \(i\), indicating complex roots.

Using the quadratic formula, we gain insight into the precise nature of the roots regardless if they are real or complex.