Problem 62 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ 4 x^{2}=x-1 $$

Step-by-Step Solution

Answer

The quadratic equation has no real solutions, only complex ones.

Step 1: Rearrange the Equation

Move all terms to one side of the equation to set it to zero. Subtract both \(x\) and \(1\) from both sides:\[ 4x^2 - x + 1 = 0 \]This rearranges the equation into a standard quadratic form \( ax^2 + bx + c = 0 \).

Step 2: Identify Coefficients

Identify the coefficients from the quadratic equation \(4x^2 - x + 1 = 0\).Here, \(a = 4\), \(b = -1\), and \(c = 1\).

Step 3: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.First, calculate the discriminant \(b^2 - 4ac\):\((-1)^2 - 4 \cdot 4 \cdot 1 = 1 - 16 = -15\).

Step 4: Interpret the Discriminant

Since the discriminant is negative (-15), the equation has no real solutions. It implies the solutions are complex numbers.

Key Concepts

Standard Form

Quadratic equations are commonly expressed in a standard form. The standard form of a quadratic equation is:

• \( ax^2 + bx + c = 0 \)

- Here, \( a \), \( b \), and \( c \) are coefficients.
The coefficient \( a \) must be non-zero. If \( a = 0 \), the equation is no longer quadratic but linear.
Rearranging any given quadratic equation into this standard form is crucial because it allows us to use tools like the quadratic formula to find its solutions.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. When a quadratic equation is in standard form, the formula becomes highly useful:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

- This formula provides solutions for \( x \) given the coefficients \( a \), \( b \), and \( c \).
- It handles both real and complex numbers.
Applying the quadratic formula correctly involves inserting the correct values for \( a \), \( b \), and \( c \) and carefully carrying out the calculations, including all operations under the square root.

Discriminant

The discriminant is a part of the quadratic formula's calculation, found under the square root:

• \( \Delta = b^2 - 4ac \)

- It is crucial because it tells us the nature of the roots of the quadratic equation.
- A positive discriminant (\( \Delta > 0 \)) indicates two distinct real roots.
- A zero discriminant (\( \Delta = 0 \)) indicates one real root, or a repeated root.
- A negative discriminant (\( \Delta < 0 \)) indicates complex solutions.
Hence, calculating \( \Delta \) helps us determine whether we have real or complex solutions without solving the entire equation.

Complex Solutions

When the discriminant is negative, the quadratic equation has complex solutions. Complex numbers occur when we take the square root of a negative number:

• A complex number has a real part and an imaginary part, expressed as \( a + bi \).
• Here, \( i \) is the imaginary unit, with \( i^2 = -1 \).

- In the context of our equation: since \( \Delta = -15 \), both solutions will involve square root of \(-15\), resulting in imaginary numbers.
Using the quadratic formula, the solutions typically appear as conjugate pairs when they are complex, maintaining symmetry in their real and imaginary parts.
Understanding complex solutions expands our ability to solve equations that do not intersect the x-axis on a real plane.