We simplify the expression under the square root and perform the multiplication in the denominator. The term b² - 4ac is called the discriminant, and its value helps determine the nature of the roots.

We simplify the expression under the square root and perform the multiplication in the denominator. The term b² - 4ac is called the discriminant, and its value helps determine the nature of the roots.

Find the roots of the quadratic equation: $3x^2 - 5x + 2 = 0$ using the quadratic formula.

Answer: The roots of the quadratic equation $3x^2 - 5x + 2 = 0$ are $x = 1$ and $x = \frac{2}{3}$ .

Step-by-step solution

Step 1: Identify coefficients

First, we compare the given quadratic equation $3x^2 - 5x + 2 = 0$ with the standard form of a quadratic equation, $ax^2 + bx + c = 0$ . This allows us to identify the coefficients $a$ , $b$ , and $c$ .

Step 2: Apply the quadratic formula

Next, we write down the quadratic formula, which is used to find the roots of any quadratic equation. This formula provides a direct way to calculate the values of $x$ .

Step 3: Substitute values into the formula

Now, we substitute the identified values of $a=3$ , $b=-5$ , and $c=2$ into the quadratic formula. This is the crucial step before performing the calculations.

Step 4: Simplify the expression

We simplify the expression under the square root and perform the multiplication in the denominator. The term $b^2 - 4ac$ is called the discriminant, and its value helps determine the nature of the roots.

Step 5: Calculate the roots

Finally, we calculate the two possible roots by considering both the positive and negative signs of the square root. This gives us the two solutions for $x$ .

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Find the roots of the quadratic equation: 3x2−5x+2=03x^2 - 5x + 2 = 03x2−5x+2=0 using the quadratic formula.