The expression on the left-side of the equation is given in the standard form of a quadratic function. Start by identifying the values of a, b, and c.
Solutions: x=- 1/2, x=- 9 Are the solutions the same? Yes.
Practice makes perfect
We want to solve the given equation for x twice — once using the Zero Product Property and once using the Quadratic Formula. Then we will compare the answers.
Using Zero Product Property
We want to solve the given equation using the Zero Product Property. To do this, we need to rewrite the given equation in the factored form. We can start by using factoring.
Factoring
Here we have a quadratic equation of the form ax^2+bx+c=0, where |a| ≠1 and there are no common factors. To factor this equation we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
2x^2+19x+9=0
We have that a= 2, b=19, and c=9. There are now three steps we need to follow in order to rewrite the above equation.
Find a c. Since we have that a= 2 and c=9, the value of a c is 2* 9=18.
Find factors of a c. Since a c=18, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=19, which is positive, those factors will need to be positive so that their sum is positive.
We will use the Quadratic Formula to solve the given quadratic equation.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
Since a= 2, b= 19, and c= 9, we will substitute these values into the Quadratic Formula.
The solutions for this equation are x= - 19± 174. Let's separate them into the positive and negative cases.
x=- 19± 17/4
x_1=- 19+17/4
x_2=- 19-17/4
x_1=- 2/4
x_2=- 36/4
x_1=- 1/2
x_2=- 9
Using the Quadratic Formula, we found that the solutions are x_1=- 12 and x_2=- 9. Notice that those are the same solutions as obtained previously by factoring and then using Zero Product Property. Both methods are equally good, so they give the same solutions.
