b The expression on the right-side of the equation is given in the standard form of a quadratic function. Start by identifying the values of a, b, and c.
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 an expression is a perfect square trinomial, we need to ask ourselves three questions.
Is the first term a perfect square?
x^2= x^2 âś“
Is the last term a perfect square?
25= 5^2 âś“
Is the middle term twice the product of 5 and x?
10x=2* 5* x âś“
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 there is a subtraction sign in the middle.
x^2-10x+25=0 ⇔ ( x- 5)^2=0
Zero Product Property
Since the equation is already written in factored form, we can now use the Zero Product Property.
Substituting and simplifying created a true statement, so we know that x=5 is a solution of the equation.
b We want to solve the given equation for x. To do this, we can use factoring. Then we will use the Zero Product Property.
Factoring
Here we have a quadratic equation of the form 0=ax^2+bx+c, 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.
0=3x^2+17x-6
⇕
0= 3x^2+17x+(- 6)
We have that a= 3, b=17, and c=- 6. There are now three steps we need to follow in order to rewrite the above equation.
Find a c. Since we have that a= 3 and c=- 6, the value of a c is 3* (- 6)=- 18.
Find factors of a c. Since ac=- 18, which is negative, factors of a c need to have opposite signs — one positive and one negative. Since b=17, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor, so that their sum is positive.
Again, we created a true statement. x=- 6 is indeed a solution of the equation.
c We want to solve the given equation for x. Before anything else, let's rewrite the equation so all terms will be gathered on one side of the equation.
3x^2-2x=5
⇕
3x^2-2x-5=0
Now, to solve the above equation for x we can start by using factoring. Then we will use the Zero Product Property.
Factoring
Here we have a quadratic equation of the form 0=ax^2+bx+c, 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.
3x^2-2x-5=0
⇕
3x^2+(- 2)x+(- 5)=0
We have that a= 3, b=- 2, and c=- 5. There are now three steps we need to follow in order to rewrite the above equation.
Find a c. Since we have that a= 3 and c=- 5, the value of a c is 3* (- 5)=- 15.
Find factors of a c. Since ac=- 15, which is negative, we need factors of a c to have opposite signs. Since b=- 2, which is also negative, the absolute value of the negative factor will need to be greater than the absolute value of the positive factor, so that their sum is negative.
Do you notice that the expression
on the left-hand side of the equation is a difference of two perfect squares? This can be factored using the difference of squares method.
a^2 - b^2 ⇔ (a+b)(a-b)
To do so, we first need to express each term as a perfect square.
Expression
16x^2-9
Rewrite as Perfect Squares
(4x)^2 - 3^2
Apply the Formula
(4x+3)(4x-3)
Solving the Equation
Finally, to solve the equation, we will use the Zero Product Property.
