Question

solve for x.
74x² - 76x + 2 = 0
write each solution as an integer, proper fraction, or improper fraction in simplest form. if there are multiple solutions, separate them with commas.
x =
save answer

Explanation:

Step1: Simplify the quadratic equation

First, we can divide the entire equation \(74x^{2}-76x + 2=0\) by 2 to simplify it. We get \(37x^{2}-38x + 1 = 0\).

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). In our simplified equation \(37x^{2}-38x + 1 = 0\), we have \(a = 37\), \(b=- 38\) and \(c = 1\).

First, calculate the discriminant \(\Delta=b^{2}-4ac\). Substitute the values: \(\Delta=(-38)^{2}-4\times37\times1=1444 - 148 = 1296\).

Then, find the square root of the discriminant: \(\sqrt{\Delta}=\sqrt{1296} = 36\).

Now, substitute into the quadratic formula:
\(x=\frac{-(-38)\pm36}{2\times37}=\frac{38\pm36}{74}\)

Step3: Calculate the two solutions

For the plus sign: \(x_{1}=\frac{38 + 36}{74}=\frac{74}{74}=1\)

For the minus sign: \(x_{2}=\frac{38-36}{74}=\frac{2}{74}=\frac{1}{37}\)

Answer:

\(1,\frac{1}{37}\)