Solve the Quadratic Equation: x² = 6x - 9 Step-by-Step
Question
Solve the following problem:
x2=6x−9
Video Solution
Solution Steps
00:00Find X
00:03Arrange the equation so the right side equals 0
00:14Factor X squared
00:17Factor 9 into 3 squared
00:21Factor 6X into factors 2, 3, and X
00:25Use the quadratic formula and find the brackets
00:32Isolate X
00:37And this is the solution to the question
Step-by-Step Solution
Proceed to solve the given equation:
x2=6x−9
First, let's arrange the equation by moving terms:
x2=6x−9x2−6x+9=0
Note that we can factor the expression on the left side by using the perfect square trinomial formula for a binomial squared:
(a−b)2=a2−2ab+b2
As shown below:
9=32Therefore, we'll represent the rightmost term as a squared term:
x2−6x+9=0↓x2−6x+32=0
Now let's examine again the perfect square trinomial formula mentioned earlier:
(a−b)2=a2−2ab+b2
And the expression on the left side in the equation that we obtained in the last step:
x2−6x+32=0
Note that the terms x2,32indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),
However, in order to factor the expression in question (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined with a single line):
(a−b)2=a2−2ab+b2
In other words - we will query whether we can represent the expression on the left side of the equation as:
x2−6x+32=0↕?x2−2⋅x⋅3+32=0
And indeed it is true that:
2⋅x⋅3=6x
Therefore we can represent the expression on the left side of the equation as a perfect square binomial:
x2−2⋅x⋅3+32=0↓(x−3)2=0
From here we can take the square root of both sides of the equation (and don't forget that there are two possibilities - positive and negative when taking an even root of both sides of an equation), then we'll easily solve by isolating the variable: