factoring to solve quadratic equation 5x^2=16x+45
5x^2 -16x -45 = 0
(5x + 9)(x -5) = 0
Either factor can be zero.
x = -9/5 or 5
To solve the quadratic equation 5x^2 = 16x + 45 using factoring, we want to rewrite the equation in the form of (x - p)(x - q) = 0, where p and q are numbers. Here's how you can approach it step-by-step:
Step 1: Move all the terms to one side of the equation to set it equal to zero:
5x^2 - 16x - 45 = 0
Step 2: Try to find two numbers, p and q, such that their sum is -16 (the coefficient of x) and their product is equal to the product of the coefficient of x^2 (5) and the constant term (-45).
In this case, we need to find two numbers whose sum is -16 and product is -225 (5 * -45). After some trial and error, we find that -20 and +4 satisfy these conditions (-20 + 4 = -16, and -20 * 4 = -80 = 5 * -45).
Step 3: Rewrite the middle term (-16x) in the equation using the values of p and q:
5x^2 - 20x + 4x - 45 = 0
Step 4: Group the terms and factor by grouping:
(5x^2 - 20x) + (4x - 45) = 0
5x(x - 4) + 1(4x - 45) = 0
Step 5: Factor out the common terms:
5x(x - 4) + 1(4x - 45) = 0
x - 4)(5x + 9) = 0
Step 6: Set each factor equal to zero:
x - 4 = 0 or 5x + 9 = 0
Step 7: Solve for x in each equation:
For x - 4 = 0, adding 4 to both sides gives:
x = 4
For 5x + 9 = 0, subtracting 9 from both sides gives:
5x = -9
Dividing both sides by 5:
x = -9/5
So, the solutions to the quadratic equation 5x^2 = 16x + 45 are x = 4 and x = -9/5.