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Chapter 17: Problem 509

Without solving, find the sum and product of the roots of \(8 x^{2}=2 x+3\)

Short Answer

Expert verified
The sum of the roots of the given quadratic equation \(8x^2 - 2x - 3 = 0\) is \(\frac{1}{4}\) and the product of the roots is \(-\frac{3}{8}\).

Step by step solution

01

Rearrange the equation into the standard quadratic form

The given equation is \(8x^2 = 2x + 3\). To rewrite it into the standard quadratic equation form, subtract both sides by 2x and 3. \[8x^2 - 2x - 3 = 0\] Now we have the equation in the standard form, \(ax^2 + bx + c = 0\), where \(a = 8\), \(b = -2\), and \(c = -3\).
02

Find the sum of the roots using Vieta's formulas

According to Vieta's formulas, the sum of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[\text{sum of the roots} = r_1 + r_2 = -\frac{b}{a}\] Using the coefficients we found in step 1, substitute their values into the formula: \[\text{sum of the roots} = -\frac{-2}{8} = \frac{1}{4}\] So, the sum of the roots of the given quadratic equation is \(\frac{1}{4}\).
03

Find the product of the roots using Vieta's formulas

According to Vieta's formulas, the product of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[\text{product of the roots} = r_1 * r_2 = \frac{c}{a}\] Using the coefficients we found in step 1, substitute their values into the formula: \[\text{product of the roots} = \frac{-3}{8}\] So, the product of the roots of the given quadratic equation is \(-\frac{3}{8}\). Without solving the equation, we found that the sum of the roots is \(\frac{1}{4}\) and the product of the roots is \(-\frac{3}{8}\).

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