what is the positive solution of the equation 5x^2 + 2x -16 =0? solve by factoring.
To solve the quadratic equation 5x^2 + 2x - 16 = 0 by factoring, we need to find two numbers whose sum is equal to the coefficient of the middle term (2x) and whose product is equal to the product of the coefficients of the first and last terms (5x^2 * -16).
Let's break down the steps to solve the equation by factoring:
Step 1: Write the equation in the form ax^2 + bx + c = 0.
For the given equation, 5x^2 + 2x - 16 = 0, we already have it in this form.
Step 2: Multiply the coefficient of the x^2 term (a) and the constant term (c) from the equation.
In this case, a = 5, and c = -16, so a * c = 5 * -16 = -80.
Step 3: Find two numbers whose sum is equal to the coefficient of the middle term (b) in the equation and whose product is equal to the result from the previous step.
The coefficient of the middle term is b = 2, and the product a * c is -80.
So, we need to find two numbers whose sum is 2 and whose product is -80.
By trial and error, we find that the two numbers are -8 and 10.
-8 + 10 = 2
-8 * 10 = -80
Step 4: Rewrite the middle term (bx) by splitting it into two terms using the numbers found in the previous step.
Replace the middle term 2x with -8x + 10x in the equation.
The equation becomes: 5x^2 - 8x + 10x - 16 = 0
Step 5: Factor by grouping.
Now, we group the first two terms and the last two terms of the equation:
(5x^2 - 8x) + (10x - 16) = 0
Factor out the greatest common factor from each group:
x(5x - 8) + 2(5x - 8) = 0
Notice that we have a common factor, (5x - 8), in both terms of the grouped equation.
Step 6: Factor out the common factor.
From the step above, we can factor out (5x - 8):
(5x - 8)(x + 2) = 0
Step 7: Set each factor equal to zero and solve for x.
We have two factors: (5x - 8) = 0 and (x + 2) = 0.
For (5x - 8) = 0:
Rearrange the equation to solve for x:
5x = 8
x = 8/5
For (x + 2) = 0:
Rearrange the equation to solve for x:
x = -2
Step 8: Check the solutions.
Plug in the values we found for x into the original equation:
For x = 8/5:
5(8/5)^2 + 2(8/5) - 16 = 0
64/5 + 16/5 - 16 = 0
64/5 + 16/5 - 80/5 = 0
(64 + 16 - 80)/5 = 0
0/5 = 0
0 = 0
For x = -2:
5(-2)^2 + 2(-2) - 16 = 0
20 - 4 - 16 = 0
20 - 20 = 0
0 = 0
Both solutions satisfy the original equation, so the positive solution of the given equation 5x^2 + 2x - 16 = 0 is x = 8/5.