Multiple choice

Find the roots of each of the following quadratic equations by the method of completing the squares $(5z + 2a) (3z + 4b) = 8ab$

  1. Roots are $\displaystyle 1 , - \frac{a + 10b}{11}$
  2. Roots are $\displaystyle 7 , - \frac{a - 15b}{17}$
  3. Roots are $\displaystyle 0 , - \frac{6a + 20b}{15}$
  4. Roots are $\displaystyle -1 , - \frac{7a - 10b}{17}$
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C Correct answer
Explanation

Expanding the equation (5z + 2a)(3z + 4b) = 8ab gives 15z^2 + 20bz + 6az + 8ab = 8ab, which simplifies to 15z^2 + (20b + 6a)z = 0. Factoring out z yields z(15z + 6a + 20b) = 0, so the roots are z = 0 and z = -(6a + 20b)/15.

AI explanation

Expanding the equation (5z plus 2a)(3z plus 4b) equals 8ab yields 15z squared plus 20bz plus 6az plus 8ab equals 8ab, which simplifies to 15z squared plus (6a plus 20b)z equals 0. Factoring out z from both terms gives z(15z plus 6a plus 20b) equals 0. Setting each factor to zero gives the first root as z equals 0, and the second root as z equals negative (6a plus 20b) divided by 15.

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