Problem 20 - Algebra 2 and Trigonometry

Question

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ 3 x^{2}=5 x-3 $$

Step-by-Step Solution

Answer

The equation has no real roots, as the discriminant is negative.

Step 1: Write the Equation in Standard Form

Start by rearranging the given equation \(3x^{2}=5x-3\) into the standard form of a quadratic equation, \(ax^2 + bx + c = 0\). Move all terms to one side: \(3x^2 - 5x + 3 = 0\). Here, \(a = 3\), \(b = -5\), and \(c = 3\).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) is given by the formula \(b^2 - 4ac\). Substitute \(b = -5\), \(a = 3\), and \(c = 3\): \[ \Delta = (-5)^2 - 4 \times 3 \times 3 = 25 - 36 = -11\]

Step 3: Interpret the Discriminant

The discriminant \(\Delta = -11\) is less than zero, which means the quadratic equation has no real roots. The roots are not real numbers, and are complex conjugates.

Step 4: Conclusion

Since the discriminant is negative, there are no real roots. The roots are not real numbers, meaning the only solutions to the equation are complex numbers.

Key Concepts

Discriminant

In quadratic equations, the discriminant serves as a vital tool to determine the nature of the roots. It is calculated using the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the equation in its standard form. By evaluating the discriminant, you can predict whether the roots are real or complex, as well as whether they are distinct or equal.

- **If the discriminant is positive** - If it is a perfect square, the equation has two real, rational, and unequal roots. - If not, the roots are real, irrational, and unequal.- **If the discriminant is zero**, the roots are real and equal, meaning they are rational and identical.- **If the discriminant is negative**, as in the example, the quadratic equation has no real roots, only complex roots. This happens because you can't take the square root of a negative number in the real number system.

Understanding the discriminant is crucial for solving quadratic equations as it gives us insights into the root types without actually solving the equation.

Complex Roots

When the discriminant is negative, the quadratic equation will have complex roots. These roots are important in mathematics because they represent solutions that are not real numbers. Complex roots come in conjugate pairs, which have the form \(a + bi\) and \(a - bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit defined by \(i^2 = -1\).

For the equation \(3x^2 - 5x + 3 = 0\), the discriminant is \(-11\). Since it is negative, the roots of this equation are complex. To find these roots, you can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Inserting the values for this equation would show roots involving \(i\), emphasizing that no real solutions exist. Understanding complex roots allows mathematicians to solve more sophisticated problems that occur in engineering and physics.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is essential for applying various solution techniques, including the calculation of the discriminant. The standard form is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.

Rewriting an equation into this form is often the first step in solving quadratic equations. It allows you to easily identify the coefficients \(a\), \(b\), and \(c\) that are necessary for using the quadratic formula or other methods to find the roots. For example, in the equation \(3x^2 = 5x -3\), rearranging it to \(3x^2 - 5x + 3 = 0\) helps facilitate further analysis.

Using the standard form ensures that you can systematically identify the nature of the roots through methods such as factoring, completing the square, or utilizing the quadratic formula. By mastering the transformation of a quadratic equation into standard form, tackling these equations becomes more efficient and less error-prone.