Ginger was asked to solve 25x^2−196=0
by factoring using the difference
of squares.
To solve this equation, we can use the difference of squares formula, which states that a^2 - b^2 can be factored as (a + b)(a - b).
In this case, we have 25x^2 - 196, which can be rewritten as (5x)^2 - 14^2.
Using the difference of squares formula, we can factor it as:
(5x + 14)(5x - 14) = 0.
Set each factor equal to zero and solve for x:
5x + 14 = 0 ; 5x - 14 = 0.
Solving these equations, we get:
5x = -14 ; 5x = 14.
Dividing both sides by 5, we find:
x = -14/5 ; x = 14/5.
Therefore, the solutions to the equation 25x^2 - 196 = 0 are x = -14/5 and x = 14/5.
To solve the equation 25x^2 - 196 = 0 by factoring using the difference, we can follow these steps:
Step 1: Recognize the equation:
We have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 25, b = 0, and c = -196.
Step 2: Apply the formula:
In this case, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b).
Step 3: Rewrite the equation:
Rewrite the equation as a difference of squares, using x^2 as a^2 and 14 as b^2:
25x^2 - 196 = (5x)^2 - 14^2
Step 4: Apply the difference of squares formula:
Apply the difference of squares formula to factor the equation:
(5x + 14)(5x - 14) = 0
Step 5: Set each factor equal to zero:
Set each factor equal to zero and solve for x:
5x + 14 = 0 or 5x - 14 = 0
Step 6: Solve for x:
Solve each equation for x:
For 5x + 14 = 0:
5x = -14
x = -14/5
For 5x - 14 = 0:
5x = 14
x = 14/5
Therefore, the solutions of the equation 25x^2 - 196 = 0 are x = -14/5 and x = 14/5.
To solve the equation 25x^2 - 196 = 0 using factoring, we can begin by recognizing that it is a quadratic equation in the form of ax^2 + bx + c = 0. In this case, a = 25, b = 0, and c = -196.
Now, let's use the difference of squares formula to factorize the equation. The difference of squares formula states that a^2 - b^2 can be factored as (a + b)(a - b).
In our equation, we have 25x^2 - 196, which can be expressed as (5x)^2 - 14^2. Therefore, we can rewrite the equation as follows:
(5x)^2 - 14^2 = 0
Now we can apply the difference of squares formula:
(5x + 14)(5x - 14) = 0
By factoring the equation using the difference of squares, we have obtained two binomial expressions set equal to zero. Now we can solve for x by setting each binomial expression equal to zero and solving for x independently.
Setting 5x + 14 = 0:
5x = -14
x = -14/5
Setting 5x - 14 = 0:
5x = 14
x = 14/5
Therefore, the solutions to the equation 25x^2 - 196 = 0 are x = -14/5 and x = 14/5.