a We want to solve the given equation for x. To do this we can start by using factoring. Then we will use the Zero Product Property.
Factoring
We want to factor the given equation. Factoring is much easier when our polynomial is a perfect square trinomial. To determine if the expression on the left-hand side of the equation is a perfect square trinomial, we need to ask ourselves three questions.
Is the first term a perfect square?
9x^2=( 3x)^2 ✓
Is the last term a perfect square?
1= 1^2 ✓
Is the middle term twice the product of 1 and 3x?
6x=2* 1* 3x ✓
As we can see, the answer to all three questions is yes! Therefore, we can write the trinomial as the square of a binomial. Note that there is a subtraction sign in the middle.
9x^2-6x+1=0 ⇔ ( 3x- 1)^2=0
Zero Product Property
Since the equation is already written in factored form, we can now use the Zero Product Property.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 aWe first need to identify the values of a, b, and c.
x^2+x+1=0 ⇕ 1x^2+ 1x+ 1=0
We see that a= 1, b= 1, and c= 1. Let's substitute these values into the Quadratic Formula.
Oops! The square root of a negative number is undefined for real numbers! As we can see, using the Quadratic Formula for the given equation resulted in contradiction. Thus, there are no real solutions.
For further explanation, notice that the left-hand side of the equation is a quadratic function. Let's graph it.
The solutions of the given equation are x-intercepts of this function. Notice that this parabola lies above the x-axis, thus it does not have x-intercepts. Therefore there are no real solutions of the given equation.
