how would i do these problems:
6. xsquared + 2x -7 =0
7. 6xsquared - 2x -1 = 0
There are 3 common ways to solve quadratic equations
1. by factoring --- only works for some
2. completing the square --- works really well if it starts with 1x^2 and the middle term is even
3. the quadratic formula --- always workd
for x^2 + 2x - 7=0 I would use "completing the square"
x^2 + 2x + 1 = 7+1
(x + 1)^2 = 8
x+1 = √8 = 2√2
x = -1 ± 2√2
for 6x2 - 2x - 1= 0 I would use the formula , which I assume you know
x = (2 ± √28)/12
= (2 ± 2√7)/12
= (1 ± √7)/6
thanks so much!
what about
4xsquared + 4x - 3= 0
done here
http://www.jiskha.com/display.cgi?id=1365552528
To solve these quadratic equations, you can use the quadratic formula or factorization method.
Let's start with problem 6:
1. Quadratic Formula Method:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, we have:
a = 1, b = 2, and c = -7.
Substituting the values into the formula, we get:
x = (-2 ± √(2^2 - 4(1)(-7))) / (2(1))
x = (-2 ± √(4 + 28)) / 2
x = (-2 ± √32) / 2
x = (-2 ± 4√2) / 2
x = -1 ± 2√2
So the solutions for this equation are:
x = -1 + 2√2 and x = -1 - 2√2.
2. Factorization Method:
To factorize the quadratic equation, we need to find two numbers that add up to 2 (the coefficient of x) and multiply to give -7 (the constant term).
The factors of -7 are -1 and 7.
Among these, the pair that adds up to 2 is -1 and 7.
So we can rewrite the equation as:
(x - 1)(x + 7) = 0
Now, set each factor equal to zero and solve for x:
x - 1 = 0 => x = 1
x + 7 = 0 => x = -7
So the solutions for this equation are:
x = 1 and x = -7.
Now let's move on to problem 7:
1. Quadratic Formula Method:
The quadratic formula is the same as mentioned earlier:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, we have:
a = 6, b = -2, and c = -1.
Substituting these values into the formula:
x = (-(-2) ± √((-2)^2 - 4(6)(-1))) / (2(6))
x = (2 ± √(4 + 24)) / 12
x = (2 ± √28) / 12
x = (2 ± 2√7) / 12
x = (1 ± √7) / 6
The solutions for this equation are:
x = (1 + √7) / 6 and x = (1 - √7) / 6.
2. Factorization Method:
To factorize the quadratic equation, we need to find two numbers whose product gives -6 (the coefficient of x^2) multiplied by -1 (the constant term).
After some trial and error, we can find that -3 and 2 satisfy this condition.
So we can rewrite the equation as:
(3x - 1)(2x + 1) = 0
Setting each factor equal to zero and solving for x:
3x - 1 = 0 => x = 1/3
2x + 1 = 0 => x = -1/2
The solutions for this equation are:
x = 1/3 and x = -1/2.
That's it! You have solved both problems using different methods.