Problem 64 - Elementary Algebra

Question

Solve using the quadratic formula. $$ t 2+3 t+4=0 $$

Step-by-Step Solution

Answer

The roots are \( t = \frac{-3 \pm i\sqrt{7}}{2} \).

Step 1: Identify the coefficients

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). For the given equation \( t^2 + 3t + 4 = 0 \), the coefficients are \( a = 1 \), \( b = 3 \), and \( c = 4 \).

Step 2: Write down the quadratic formula

The quadratic formula to find the roots of \( ax^2 + bx + c = 0 \) is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 3: Calculate the discriminant

The discriminant \( \Delta \) is given by \( b^2 - 4ac \). Substituting the values, we have \( \Delta = 3^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7 \).

Step 4: Determine the nature of the roots

Since the discriminant \( \Delta = -7 \) is negative, the roots of the equation are complex (non-real numbers).

Step 5: Apply the quadratic formula

Substituting the values into the quadratic formula yields: \( t = \frac{-3 \pm \sqrt{-7}}{2 \cdot 1} = \frac{-3 \pm i\sqrt{7}}{2} \).

Step 6: Simplify the roots

The roots can be written separately as \( t_1 = \frac{-3 + i\sqrt{7}}{2} \) and \( t_2 = \frac{-3 - i\sqrt{7}}{2} \).

Key Concepts

Discriminant

The discriminant is a crucial part of the quadratic formula. It helps determine the type and number of roots that a quadratic equation will have. The discriminant is represented by the expression \( \Delta = b^2 - 4ac \). This expression is derived from the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Understanding the discriminant is important because:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, which means the roots are repeated or double.
• If \( \Delta < 0 \), as seen in our exercise with \( \Delta = -7 \), the quadratic equation has two complex roots, meaning the roots are not real numbers.

By calculating the discriminant before solving the equation, you can predict the nature of the roots quickly and effectively.

Complex Roots

When the discriminant is negative, the roots of the quadratic equation are complex. Unlike real roots, complex roots involve imaginary numbers.

Imaginary numbers come into play because you will have a negative number under the square root when calculating the roots using the quadratic formula.
Imaginary numbers are expressed using \( i \), which is the square root of \(-1\).

In the original exercise, the discriminant \( \Delta = -7 \). This results in complex roots, which are calculated as follows:

• The expression under the square root becomes \( \sqrt{-7} \)
• Using \( i \), this turns into \( i\sqrt{7} \)
• Thus, the roots are \( t = \frac{-3 \pm i\sqrt{7}}{2} \)

Complex roots usually come in conjugate pairs, which means if one root is \( \frac{-3 + i\sqrt{7}}{2} \), the other is \( \frac{-3 - i\sqrt{7}}{2} \).This pairing helps maintain the symmetry typical of quadratic functions.

Coefficients of Quadratic Equation

The coefficients are the constants \( a \), \( b \), and \( c \) in the standard form of a quadratic equation, \( ax^2 + bx + c = 0 \). They play a vital role in both identifying the shape of the graph of the quadratic function and solving the equation.

• \( a \): The coefficient of \( t^2 \) (or \( x^2 \)) is the leading coefficient. It dictates the parabola's width and direction (upward if \( a > 0 \), downward if \( a < 0 \)).
• \( b \): The coefficient of \( t \) (or \( x \)), which influences the axis of symmetry and the parabola's vertex.
• \( c \): This is the constant term, representing the point where the parabola intersects the y-axis (in function terms).

In the quadratic equation from the exercise, \( t^2 + 3t + 4 = 0 \), the coefficients are:

• \( a = 1 \)
• \( b = 3 \)
• \( c = 4 \)

Knowing these coefficients is the first step in using the quadratic formula effectively, ensuring accurate calculation of the roots, whether real or complex.