Questions: Find the real solutions, if any, of the following equation. Use the quadratic formula. 7 x^2 = 2 - 9 x Select the correct choice below and fill in any answer boxes within your choice. A. The solution set is . (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. There are no solutions.

Solution Steps

To find the real solutions of the quadratic equation \(7x^2 = 2 - 9x\), we first need to rewrite it in the standard form \(ax^2 + bx + c = 0\). Then, we can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions. The discriminant \(b^2 - 4ac\) will help us determine the nature of the roots (real or complex).

The given equation is \(7x^2 = 2 - 9x\). To rewrite it in the standard quadratic form \(ax^2 + bx + c = 0\), we move all terms to one side:

\[ 7x^2 + 9x - 2 = 0 \]

From the equation \(7x^2 + 9x - 2 = 0\), we identify the coefficients:

• \(a = 7\)
• \(b = 9\)
• \(c = -2\)

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\[ \Delta = b^2 - 4ac \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ \Delta = 9^2 - 4 \times 7 \times (-2) = 81 + 56 = 137 \]

Since the discriminant \(\Delta = 137\) is positive, the quadratic equation has two distinct real solutions.

The solutions of the quadratic equation are given by the quadratic formula:

\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]

Substituting the values of \(b\), \(\Delta\), and \(a\):

\[ x = \frac{-9 \pm \sqrt{137}}{14} \]

Final Answer