Question

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ 4 u^{2}-6 u+9=0 $$

Short answer

The quadratic equation has no real solutions because the discriminant is negative.

Step-by-step solution

Identify Coefficients

Identify the coefficients in the quadratic equation \[ 4u^2 - 6u + 9 = 0 \] The coefficients are: \[ a = 4 \] \[ b = -6 \] \[ c = 9 \]

Quadratic Formula

Write down the quadratic formula \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Calculate the Discriminant

Calculate the discriminant \[ D = b^2 - 4ac \] Substitute the values for a, b, and c: \[ D = (-6)^2 - 4(4)(9) = 36 - 144 = -108 \]

Analyze the Discriminant

Since the discriminant \[ D = -108 \] is less than 0, the quadratic equation has no real solutions.

Key concepts

Real Solutions

In mathematics, real solutions are the values of the variable that satisfy a given equation and are real numbers (not imaginary or complex). When dealing with quadratic equations, we use the quadratic formula to find the solutions. A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \).

Generally, a quadratic equation can have:

• Two real solutions
• One real solution
• No real solutions (if the solutions are complex)

To determine the nature of the solutions, we look at the discriminant (\( D \)).
This is essential in figuring out whether the solutions are real or complex.
In the example equation \( 4u^2 - 6u + 9 = 0 \), analyzing the discriminant reveals there are no real solutions.

Discriminant

The discriminant is a specific part of the quadratic formula that helps determine the nature of the solutions to the quadratic equation. It is found inside the square root in the quadratic formula: \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
The discriminant is represented by \( D \) and given by the formula: \( D = b^2 - 4ac \).

The value of the discriminant tells us the type of solutions:

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution.
• If \( D < 0 \), there are no real solutions; instead, there are two complex solutions.

In the equation \( 4u^2 - 6u + 9 = 0 \), the discriminant is calculated as \( D = (-6)^2 - 4(4)(9) = 36 - 144 = -108 \).
Since \( D < 0 \), this confirms that the equation has no real solutions.

Coefficients

Coefficients in a quadratic equation are the numerical factors that multiply the variables. They are essential in determining the properties and solutions of the equation. In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are:

• \( a \): the coefficient of \( x^2 \)
• \( b \): the coefficient of \( x \)
• \( c \): the constant term or coefficient of \( x^0 \)

Identifying the coefficients is the first step in using the quadratic formula.
For the example \( 4u^2 - 6u + 9 = 0 \), we have:

• \( a = 4 \)
• \( b = -6 \)
• \( c = 9 \)

These coefficients are crucial for determining the discriminant and ultimately solving the quadratic equation.
By substituting \( a, b, \) and \( c \) into the formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we can find the solutions step-by-step.