Solve the following quadratic equation using the perfect square trinomial pattern: 49x2=−392x−784
.(1 point)
Responses
x=4
x equals 4
x=−784
x equals negative 784
x=−4
x equals negative 4
x=784
x equals 784
To solve the quadratic equation using the perfect square trinomial pattern, follow these steps:
1. Write the equation in the form of ax^2 + bx + c = 0. In this case, the equation is:
49x^2 + 392x + 784 = 0
2. Look at the coefficient of x^2, which is 49. Take the square root of this coefficient. √49 = 7. This will be the coefficient of x in the perfect square trinomial pattern.
3. Next, divide the coefficient of x by 2, and square it. In this case, 392/2 = 196. So, 196^2 = 38416. This will be the constant term in the perfect square trinomial pattern.
4. Rewrite the equation by adding and subtracting the constant term obtained in the previous step. The equation becomes:
49x^2 + 392x + 38416 - 38416 + 784 = 0
5. Rearrange the equation by grouping the terms:
(49x^2 + 392x + 38416) - 38416 + 784 = 0
6. Simplify the grouped terms:
(7x + 196)^2 - 37632 + 784 = 0
7. Combine like terms:
(7x + 196)^2 - 36848 = 0
8. Move the constant term to the other side of the equation:
(7x + 196)^2 = 36848
9. Take the square root of both sides of the equation:
7x + 196 = ±√36848
10. Simplify the square root:
7x + 196 = ±√(16 * 2303)
7x + 196 = ±4√2303
11. Solve for x by subtracting 196 and dividing by 7:
a) 7x = -196 + 4√2303
x = (-196 + 4√2303)/7
b) 7x = -196 - 4√2303
x = (-196 - 4√2303)/7
So, the solutions to the quadratic equation are:
x = (-196 + 4√2303)/7
x = (-196 - 4√2303)/7
not the right answer
I apologize for the confusion. Let's try solving the equation again using the correct method.
The equation is:
49x^2 = -392x - 784
Step 1: Write the equation in the form of ax^2 + bx + c = 0:
49x^2 + 392x + 784 = 0
Step 2: Divide the equation by the leading coefficient 49:
x^2 + 8x + 16 = 0
Step 3: Recognize that the equation can be factored as a perfect square trinomial:
(x + 4)^2 = 0
Step 4: Set each factor equal to zero and solve for x:
x + 4 = 0
x = -4
The solution to the quadratic equation is x = -4.
To solve the quadratic equation 49x^2 = -392x - 784 using the perfect square trinomial pattern, we can first move all terms to one side to get:
49x^2 + 392x + 784 = 0
Next, we can try to rewrite the left side of the equation as a perfect square trinomial. The perfect square trinomial pattern is (a + b)^2 = a^2 + 2ab + b^2.
In this case, we have:
(7x + 28)^2 = 49x^2 + 2(7x)(28) + 28^2
= 49x^2 + 392x + 784
Since the left side is equal to the original equation, we can rewrite the equation as:
(7x + 28)^2 = 0
Now, we can take the square root of both sides to solve for x:
7x + 28 = 0
7x = -28
x = -28/7
x = -4
Therefore, the solution to the quadratic equation is x = -4.