Question

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ 4 y^{2}-y+2=0 $$

Short answer

There are no real solutions for the equation 4y^2 - y + 2 = 0.

Step-by-step solution

Identify the coefficients

To apply the quadratic formula, identify the coefficients of the given quadratic equation. The standard form of a quadratic equation is y = ax^2 + bx + cFor the given equation 4y^2 - y + 2 = 0,the coefficients are: a = 4,b = -1,and c = 2.

Write the quadratic formula

The quadratic formula to find the roots of a quadratic equation ax^2 + bx + c = 0is given by:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substitute the coefficients into the quadratic formula

Substitute the identified coefficients a = 4, b = -1, and c = 2 into the quadratic formula:\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(2)}}{2(4)} \]which simplifies to\[ y = \frac{1 \pm \sqrt{1 - 32}}{8} = \frac{1 \pm \sqrt {-31}}{8} \]

Simplify the expression under the square root

Calculate the value under the square root:\[-31 \]Since the value under the square root is negative (-31), the solutions are complex and not real.

State the conclusion

Since the discriminant (the value under the square root) is negative, there are no real solutions for the equation 4y^2 - y + 2 = 0.The solutions are complex.

Key concepts

quadratic equation

A quadratic equation is a polynomial equation of the second degree. It has the general form: \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

In our problem, the quadratic equation is: \( 4y^2 - y + 2 = 0 \).
Here, the coefficients are \( a = 4 \), \( b = -1 \), and \( c = 2 \). These coefficients are crucial for solving the equation using the quadratic formula.

real solutions

Real solutions to a quadratic equation are the values of \(x\) that satisfy the equation and are real numbers. To determine if a quadratic equation has real solutions, we use the quadratic formula: \( x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \).

A critical part of the formula is the discriminant, \( b^2 - 4ac \). If the discriminant is:

• Positive: The equation has two distinct real solutions.
• Zero: The equation has exactly one real solution.
• Negative: The equation has no real solutions (but has complex solutions).

complex numbers

Complex numbers are numbers that include the imaginary unit \( i \), where \( i^2 = -1 \). They have the form \( a + bi \), where \( a \) and \( b \) are real numbers.

In the context of quadratic equations, if the discriminant is negative, the solutions are not real but complex.

For our problem, the discriminant is \( -31 \), leading to complex solutions: \( y = \frac{1 \, \pm \, \sqrt{-31}}{8} \) which can be expressed using \( i \) as: \( y = \frac{1}{8} \, \pm \, \frac{\sqrt{31} i}{8} \). These are the two complex solutions.

discriminant

The discriminant of a quadratic equation is calculated using the formula: \( \text{Discriminant} = b^2 - 4ac \).

It helps to determine the nature of the solutions for a quadratic equation.

For our given quadratic equation \( 4y^2 - y + 2 = 0 \):

• \( b^2 = (-1)^2 = 1 \)
• \( 4ac = 4(4)(2) = 32 \)
• \( b^2 - 4ac = 1 - 32 = -31 \)
  
  Since the discriminant is \( -31 \), a negative number, our quadratic equation has no real solutions but has two complex solutions.