I am having trouble with simple algebra. It has been 27 years since I last encountered any real math, and I am in the process of reviewing to help my grandson. Can I please get step-by step instructions on how to solve this simple math problem?:
x^2+5x+9=5
Any help is appreciated.
an equation of this type is called a "quadratic equation." there are three main ways to solve it.Originally Posted by Emodius
1) By the quadratic formula ---will work in any case
2) By completing the square ---will work in any case
3) By foiling ---works with "nice" quadratics, but is generally simpler so use when you can
which method is your grandson currently learning?
Three methods:
1. Factoring
x^2 + 5x + 9 = 5
x^2 + 5x + 4 = 0
Multiply the coefficient of the x^2 term (1) and the constant term (4): 1 x 4 = 4
Now write out all the factors of 4:
1, 4 <-- This also covers 4, 1
2, 2
-1, -4
-2, -2
Now look for the pair that adds to the coefficient of the linear term (5):
1 + 4 = 5
2 + 2 = 4
-1 + -4 = -5
-2 + -2 = -4
So we need the 1, 4 pair.
Now write the equation as the following:
x^2 + 5x + 4 = 0
x^2 + (1x + 4x) + 4 = 0
(x^2 + x) + (4x + 4) = 0
Now factor the common factor from each set of parenthesis:
x(x + 1) + 4(x + 1) = 0
Now each term has a common factor, the x + 1. So factor this from the overall expression:
(x + 4)(x + 1) = 0
Now, if this is to be true, one of (or both of) the factors must be 0.
Thus x + 4 = 0 ==> x = -4
or
x + 1 = 0 ==> x = -1
Thus the solution is x = -4, -1.
(This method works for any problem that factors. If it doesn't factor then you won't be able to find a pair of numbers that add to be the coefficient of the linear term. In that case you need to use one of the other two techniques.)
-Dan
2. Completing the square.
x^2 + 5x + 9 = 5
x^2 + 5x = -4
Compare the LHS of the equation with the LHS of the following:
x^2 + 2ax + a^2 = (x + a)^2
We see that if we find the right a value then all we need to do is add a^2 to both sides to make the LHS a perfect square.
So compare the two linear coefficients:
5 = 2a ==> a = 5/2
Thus we need to add a^2 = 25/4 to both sides of the equation:
x^2 + 5x = -4
x^2 + 5x + 25/4 = -4 + 25/4
(x + 5/2)^2 = -4 + 25/4 = -16/4 + 25/4 = 9/4
(x + 5/2)^2 = 9/4
x + 5/2 = (+/-)sqrt{9/4} = (+/-)(3/2)
x = -5/2 (+/-) 3/2
So
x = -5/2 + 3/2 = -2/2 = -1
or
x = -5/2 - 3/2 = -8/2 = -4
-Dan
3. The quadratic formula
x^2 + 5x + 4 = 0
If we have the trinomial ax^2 + bx + c = 0 then
x = [-b (+/-) sqrt{b^2 - 4ac}]/[2a]
In this case a = 1, b = 5, c = 4.
x = [-5 (+/-) sqrt{5^2 - 4*1*4}]/[2*1]
x = [-5 (+/-) sqrt{25 - 16}]/2
x = [-5 (+/-) sqrt{9}]/2
x = [-5 (+/-) 3]/2
So
x = [-5 + 3]/2 = [-2]/2 = -1
or
x = [-5 - 3]/2 = [-8]/2 = -4
as you might have expected by now.
-Dan
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