For the : ax^2 + bx + c
How would you get what to put in
(_+_)(_+_)= 0
Is it c doubled and then finding the factors that could equal b or 1/2 of b times c...?
Could someone post how I could find the answer this way (shown above)
(AND NOT THE QUADRATIC FORMULA)?
Thnak You.
ax^2 + bx + c
is NOT an equation.
Why should you expect to be able to rewrite it as (_+_)(_+_)= 0 ?
The values of x that satisfy the equation
ax^2 + bx + c = 0
ARE given by the quadratic formula.
Call them x1 and x2.
(x-x1)(x-x2) = 0
x1 = (1/2a)[-b + sqrt(b^2-4ac)]
x2 = (1/2a)[-b - sqrt(b^2-4ac)]
a ( x^2 + (b/a) x + (c/a))
a ( x^2 + 2(b/(2a)) x + (c/a))
a [ x^2 + 2(b/(2a)) x + (b/(2a))^2 +
- ( (b/(2a))^2 - (c/a) )]
a [( x + (b/(2a)))^2 -
- ( (b/(2a))^2 - (c/a) )]
a [( x + (b/(2a)))^2 -
- ( (b^2 - 4ac/((2a)^2)]
a [( x + (b/(2a)))^2 -
- ( (b^2 - 4ac/((2a)^2)]
a [( x + (b/(2a))) - ( sqrt(b^2 - 4ac/((2a))] *
[( x + (b/(2a))) + ( sqrt(b^2 - 4ac/((2a))]
a [( x + ((b/(2a)) - sqrt(b^2 - 4ac/((2a))] *
[( x + (b/(2a))) + sqrt(b^2 - 4ac/((2a))]
To determine what goes into the parentheses of (_+_)(_+_)=0 for the quadratic equation ax^2 + bx + c = 0, you can use a technique called factoring. Factoring involves finding two numbers that multiply to give you the constant term (c) and add or subtract to give you the coefficient of the middle term (b).
Here's how you can find the factors:
Step 1: Multiply the coefficient of the x^2 term (a) by the constant term (c). Let's call this product ac.
Step 2: Find two numbers that multiply to give you ac and add up to give you the coefficient of the x term (b). These two numbers will be the factors you are looking for.
Step 3: Once you have determined the two numbers, substitute them into the parentheses in the expression (_+_)(_+_) = 0. Typically, you would place the larger factor in the first set of parentheses and the smaller factor in the second set of parentheses.
Here's an example to illustrate the process:
Let's say we have the quadratic equation 2x^2 + 5x + 2 = 0.
Step 1: Multiply the coefficient of the x^2 term (2) by the constant term (2): 2 * 2 = 4 (ac = 4).
Step 2: Find two numbers that multiply to give you 4 and add up to give you the coefficient of the x term (5). In this case, 4 and 1 meet this criterion.
Step 3: Substitute these numbers into the parentheses: (2x + 4)(x + 1) = 0.
By setting each factor within the parentheses equal to zero, you can then solve for x. In this example, the solutions are x = -2 and x = -1.
Using factoring in this way allows you to solve quadratic equations by finding the factors that satisfy the equation (_+_)(_+_) = 0 without resorting to the quadratic formula.