Find the roots of the quadratic equation 3x2−26x+2=0.
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Step-by-Step Solution
Step 1: Identify coefficients
We identify the coefficients a, b, and c from the given quadratic equation 3x2−26x+2=0 by comparing it with the standard form of a quadratic equation, ax2+bx+c=0.
Step 2: Calculate the Discriminant
First, we calculate the discriminant D using the formula D=b2−4ac. This value will determine the nature of the roots. Here, a=3, b=−26, and c=2. We substitute these values into the formula.
Step 3: Determine the nature of roots
Since the discriminant D=0, the quadratic equation has two equal real roots. This is an important property that simplifies finding the roots, as mentioned in the NCERT textbook (reference 1, 3).
Step 4: Apply the quadratic formula
To find the roots, we use the quadratic formula, which is x=2a−b±b2−4ac. Since we already calculated b2−4ac=0, this simplifies the formula significantly.
Step 5: Substitute values and simplify
We substitute the values of a, b, and the discriminant D=0 into the quadratic formula. This leads to a single value for x, representing the two equal real roots. We then simplify the expression to get the final root.
Step 6: Express the roots
Since the discriminant was zero, the quadratic equation has two equal real roots. Therefore, both roots are 36.