First, we compare the given quadratic equation 2x² + 5x - 3 = 0 with the standard form of a quadratic equation, ax² + bx + c = 0. This allows us to identify the values of the coefficients a, b, and c.

First, we compare the given quadratic equation 2x² + 5x - 3 = 0 with the standard form of a quadratic equation, ax² + bx + c = 0. This allows us to identify the values of the coefficients a, b, and c.

First, calculate the term inside the square root, which is b² - 4ac. Here, 5² = 25 and 4(2)(-3) = -24. Subtracting a negative number is equivalent to adding a positive number.

The square root of 49 is 7. Now, we can replace √49 with 7 in the formula. This will give us two possible solutions for x.

Solve using the quadratic formula: 2x² + 5x − 3 = 0.

Answer: The solutions are $x = \frac{1}{2}$ and $x = -3$ .

Step-by-step solution

Step 1: Identify coefficients

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

Step 2: Apply the quadratic formula

Now that we have the values of $a$ , $b$ , and $c$ , we can substitute them into the quadratic formula. The quadratic formula is used to find the roots of any quadratic equation.

Step 3: Substitute values into the formula

Substitute the identified values $a=2$ , $b=5$ , and $c=-3$ into the quadratic formula. Be careful with the signs, especially when multiplying negative numbers.

Step 4: Simplify the expression under the square root

First, calculate the term inside the square root, which is $b^2 - 4ac$ . Here, $5^2 = 25$ and $4(2)(-3) = -24$ . Subtracting a negative number is equivalent to adding a positive number.

Step 5: Calculate the square root and simplify

The square root of 49 is 7. Now, we can replace $\sqrt{49}$ with 7 in the formula. This will give us two possible solutions for $x$ .

Step 6: Find the two solutions for x

We now separate the expression into two cases: one with the plus sign and one with the minus sign. This gives us the two distinct roots of the quadratic equation.

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