Find the real solutions, if any, of the following equation. Use the quadratic formula.

Select the correct choice below and, if necessary, fill in the answer box to complete your answer.
A. The solution set is
(Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate answers
as needed. Use integers or fractions for any numbers in the expression.)
B. The solution is not a real number.

Question

Find the real solutions, if any, of the following equation. Use the quadratic formula.

Select the correct choice below and, if necessary, fill in the answer box to complete your answer.
A. The solution set is

(Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate answers
as needed. Use integers or fractions for any numbers in the expression.)
B. The solution is not a real number.

UpStudy AI · Accepted Answer

To solve the equation $$ 5x^{2} = 6x $$ using the quadratic formula, follow these steps:

1. **Rewrite the Equation in Standard Form:**
   
   Bring all terms to one side to set the equation to zero:
   $$
   5x^{2} - 6x = 0
   $$
   
   This is in the standard quadratic form $$ ax^{2} + bx + c = 0 $$, where:
   $$
   a = 5,\quad b = -6,\quad c = 0
   $$

2. **Apply the Quadratic Formula:**
   
   The quadratic formula is:
   $$
   x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
   $$
   
   Plugging in the values:
   $$
   x = \frac{-(-6) \pm \sqrt{(-6)^{2} - 4 \cdot 5 \cdot 0}}{2 \cdot 5}
   $$
   $$
   x = \frac{6 \pm \sqrt{36}}{10}
   $$
   $$
   x = \frac{6 \pm 6}{10}
   $$
   
3. **Calculate the Solutions:**
   
   - **First Solution:**
     $$
     x = \frac{6 + 6}{10} = \frac{12}{10} = \frac{6}{5}
     $$
   
   - **Second Solution:**
     $$
     x = \frac{6 - 6}{10} = \frac{0}{10} = 0
     $$
   
   Both solutions are real numbers.

**Final Answer:**

**A.** The solution set is $$ \left\{ 0,\, \dfrac{6}{5} \right\} $$
The solution set is $$ \left\{ 0,\, \dfrac{6}{5} \right\} $$.

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