To solve the equation
\[ 4(5x - 10)^2 - 34 = -33, \]
let's start by isolating the quadratic term. We can add 34 to both sides:
\[ 4(5x - 10)^2 - 34 + 34 = -33 + 34, \]
which simplifies to
\[ 4(5x - 10)^2 = 1. \]
Next, we divide both sides by 4:
\[ (5x - 10)^2 = \frac{1}{4}. \]
Now, we take the square root of both sides. Remember to consider both the positive and negative roots:
\[ 5x - 10 = \frac{1}{2} \quad \text{or} \quad 5x - 10 = -\frac{1}{2}. \]
Case 1:
\[ 5x - 10 = \frac{1}{2} \]
Adding 10 to both sides gives:
\[ 5x = \frac{1}{2} + 10. \]
Converting 10 to a fraction:
\[ 10 = \frac{20}{2} \Rightarrow 5x = \frac{1}{2} + \frac{20}{2} = \frac{21}{2}. \]
Now, divide both sides by 5:
\[ x = \frac{21/2}{5} = \frac{21}{10}. \]
Case 2:
\[ 5x - 10 = -\frac{1}{2}. \]
Adding 10 to both sides gives:
\[ 5x = -\frac{1}{2} + 10. \]
Converting 10 to a fraction:
\[ 10 = \frac{20}{2} \Rightarrow 5x = -\frac{1}{2} + \frac{20}{2} = \frac{19}{2}. \]
Now divide both sides by 5:
\[ x = \frac{19/2}{5} = \frac{19}{10}. \]
Thus, the solutions for the quadratic equation are:
\[ x = \frac{21}{10} \quad \text{and} \quad x = \frac{19}{10}. \]
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