2a^2+5a-3=0 solve by factoring

What is

(2a-1)(a+3)

To solve the quadratic equation 2a^2 + 5a - 3 = 0 by factoring, we need to factorize the equation into two binomial expressions set equal to zero. Here's how you can do it step by step:

Step 1: Write down the equation: 2a^2 + 5a - 3 = 0.

Step 2: Multiply the coefficient of 'a^2' (2) with the constant term (-3). In this case, 2 * -3 equals -6.

Step 3: Find two numbers that multiply to give -6 and add up to the coefficient of 'a' (5). In this case, the numbers are 6 and -1 because 6 * -1 equals -6 and 6 + (-1) equals 5.

Step 4: Rewrite the middle term (5a) as the sum of the two numbers found in step 3. The equation now becomes: 2a^2 + 6a - a - 3 = 0.

Step 5: Group the terms in pairs and factor out the greatest common factor from each pair. The equation becomes: (2a^2 + 6a) + (-a - 3) = 0.

Step 6: Factor out the common factors. It becomes: 2a(a + 3) - 1(a + 3) = 0.

Step 7: Notice that we have a common factor, (a + 3), in both terms. Factor it out: (2a - 1)(a + 3) = 0.

Step 8: Now, equate each factor to zero by applying the zero product property. Set (2a - 1) = 0 and (a + 3) = 0.

Step 9: Solve for 'a' in each equation. For (2a - 1) = 0, add 1 to both sides and divide by 2 to get a = 1/2. For (a + 3) = 0, subtract 3 from both sides to get a = -3.

So, the solutions to the equation 2a^2 + 5a - 3 = 0 are a = 1/2 and a = -3.