Question

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ 2 x^{2}-5 x+3=0 $$

Short answer

The solutions are \( x = 1.5 \) and \( x = 1 \).

Step-by-step solution

- Identify coefficients

For the quadratic equation in the form \[ ax^2 + bx + c = 0 \], identify coefficients. Here, \( a = 2 \), \( b = -5 \), \( c = 3 \).

- Write the Quadratic Formula

The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

- Calculate the Discriminant

Calculate the discriminant using \( b^2 - 4ac \). With \( b = -5 \), \( a = 2 \) and \( c = 3 \), compute:\[ (-5)^2 - 4(2)(3) = 25 - 24 = 1 \].

- Apply the Quadratic Formula

Substitute the values into the quadratic formula:\[ x = \frac{-(-5) \pm \sqrt{1}}{2(2)} = \frac{5 \pm 1}{4} \].

- Simplify

Solve for the two possible values of \( x \):\[ x = \frac{5 + 1}{4} = \frac{6}{4} = 1.5 \]\[ x = \frac{5 - 1}{4} = \frac{4}{4} = 1 \].

Key concepts

quadratic equation

A quadratic equation is a type of polynomial equation of degree 2. This means the highest exponent of a variable in the equation is 2. Typically, it is written in the standard form: i.e., \[ ax^2 + bx + c = 0 \]where:

• a is the coefficient of x^2 and cannot be zero.
• b is the coefficient of x.
• c is the constant term.

Understanding the structure of a quadratic equation helps in identifying the necessary coefficients when solving it using different methods such as factoring, completing the square, or using the quadratic formula.

discriminant

The discriminant is a key part of the quadratic formula and plays an essential role in determining the nature of the solutions of a quadratic equation. It is found in the part under the square root (radical) in the quadratic formula. The discriminant (D) is calculated as: \[ D = b^2 - 4ac \] Where:

• b is the coefficient of x.
• a is the coefficient of x².
• c is the constant term.

The value of the discriminant tells us about the roots of the quadratic equation. If:

• D > 0: The equation has two distinct real solutions.
• D = 0: The equation has exactly one real solution (also called a repeated root).
• D < 0: The equation has no real solutions (the solutions are complex or imaginary numbers).

real solutions

When solving quadratic equations, we often look for real solutions—these are the x-values that satisfy the equation and are real numbers (not imaginary). Using the discriminant calculated from the quadratic formula, we can determine if there are any real solutions:

1. Two distinct real solutions: This occurs when the discriminant (D) is greater than zero.
2. One real solution: This occurs when the discriminant (D) equals zero. In this special case, the quadratic equation has a repeated (or double) root.
3. No real solutions: This occurs when the discriminant (D) is less than zero. Here, the roots are complex (imaginary) numbers.

In our exercise, the discriminant was calculated as 1 (D = 1), implying our equation has two distinct real solutions.

solving equations

To solve quadratic equations, one of the most effective methods is using the quadratic formula. This formula provides a straightforward way to find the roots of any quadratic equation. Here’s a step-by-step guide that was applied in our exercise:

• Identify coefficients: Extract the values of a, b, and c from the standard form of the equation \[ax^2 + bx + c = 0.\] In our example, a = 2, b = -5, and c = 3.
• Write the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• Calculate the discriminant: Use the coefficients to calculate the discriminant \[ D = b^2 - 4ac.\] For our equation, \[ D = (-5)^2 - 4(2)(3) = 1.\]
• Apply the quadratic formula: Substitute the values into the formula to find x: \[ x = \frac{-(-5) \pm \sqrt{1}}{2(2)}.\]
• Simplify the solutions: Solve for the two potential values of x. Here, \[ x = 1.5\] and \[ x = 1.\]

This process is a reliable method for finding the roots of any quadratic equation, ensuring no solution is overlooked.