Solve the equation for real solutions by the quadratic formula.
2x2=7x+1
To solve the equation 2x^2 = 7x + 1 using the quadratic formula, we first rewrite it in standard form:
2x^2 - 7x - 1 = 0
Now we can identify the coefficients a, b, and c in the general quadratic equation ax^2 + bx + c = 0:
a = 2
b = -7
c = -1
The quadratic formula states that the solutions for x can be found using the formula:
x = [-b ± √(b^2 - 4ac)] / 2a
Plugging in the values from our equation gives:
x = [7 ± √((-7)^2 - 4(2)(-1))] / (2)(2)
x = [7 ± √(49 + 8)] / 4
x = [7 ± √57] / 4
Therefore, the solutions for x are:
x = (7 + √57) / 4
and
x = (7 - √57) / 4
To solve the equation 2x^2 = 7x + 1 using the quadratic formula, we first need to rearrange the equation in the standard quadratic form, which is ax^2 + bx + c = 0.
In our equation, 2x^2 = 7x + 1, we have a coefficient of 2 for x^2 term, -7x as the coefficient for the x term, and 1 as the constant term.
So, the equation can be rewritten as:
2x^2 - 7x - 1 = 0.
Now, we can directly plug the values into the quadratic formula, which is:
x = (-b ± √(b^2 - 4ac)) / 2a.
In our equation, a = 2, b = -7, and c = -1.
Let's substitute these values into the quadratic formula:
x = (-(-7) ± √((-7)^2 - 4(2)(-1))) / (2(2)).
Simplifying further:
x = (7 ± √(49 + 8)) / 4.
x = (7 ± √(57)) / 4.
Hence, the solutions to the equation 2x^2 = 7x + 1, using the quadratic formula, are:
x = (7 + √57) / 4, and
x = (7 - √57) / 4.
To solve the given equation using the quadratic formula, we need to make sure it is in the form of "ax^2 + bx + c = 0".
Given equation: 2x^2 = 7x + 1
To get the equation in the desired form, we'll move all the terms to one side:
2x^2 - 7x - 1 = 0
Now we can identify the coefficients:
a = 2
b = -7
c = -1
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the known coefficients, we have:
x = (-(-7) ± √((-7)^2 - 4(2)(-1))) / (2(2))
Simplifying further:
x = (7 ± √(49 + 8)) / 4
x = (7 ± √57) / 4
Hence, the real solutions to the equation 2x^2 = 7x + 1 are:
x = (7 + √57) / 4
and
x = (7 - √57) / 4