Today we will solve a quadratic equation using the quadratic formula. Our equation is 2x squared plus 5x minus 3 equals zero. We identify the coefficients: a equals 2, b equals 5, and c equals negative 3.
Now we calculate the discriminant. Delta equals b squared minus 4ac. Substituting our values: Delta equals 5 squared minus 4 times 2 times negative 3. This gives us 25 minus negative 24, which equals 25 plus 24, equals 49. Since the discriminant is positive, we have two distinct real roots.
Now we apply the quadratic formula. Substituting our values: x equals negative 5 plus or minus square root of 49, all over 4. Since square root of 49 equals 7, we get x equals negative 5 plus or minus 7, over 4. For the first root: x equals negative 5 plus 7 over 4, which equals 2 over 4, or one half. For the second root: x equals negative 5 minus 7 over 4, which equals negative 12 over 4, or negative 3.
Now let's verify our solutions. For x equals one half: substituting into the original equation gives us 2 times one quarter plus 5 times one half minus 3, which equals one half plus five halves minus 3, equals zero. Check! For x equals negative 3: substituting gives us 2 times 9 plus 5 times negative 3 minus 3, which equals 18 minus 15 minus 3, equals zero. Both solutions are correct!