Solve using the quadratic formula: 2x² + 5x − 3 = 0.
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Step-by-Step Solution
Step 1: Identify coefficients
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.
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 b2−4ac. Here, 52=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 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.