Questions: Solve the equation using the quadratic formula. 2 x(x-3)=7 x+7 The solution set is . (Simplify your answer, including any radicals and i as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

Solution Steps

To solve the quadratic equation \(2x(x-3) = 7x + 7\), first expand and rearrange the equation into the standard quadratic form \(ax^2 + bx + c = 0\). Then, apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(x\).

Starting with the equation \(2x(x-3) = 7x + 7\), we first expand and rearrange it into standard quadratic form: \[ 2x^2 - 6x - 7x - 7 = 0 \implies 2x^2 - 13x - 7 = 0 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we identify the coefficients:

• \(a = 2\)
• \(b = -13\)
• \(c = -7\)

Calculating the discriminant: \[ b^2 - 4ac = (-13)^2 - 4 \cdot 2 \cdot (-7) = 169 + 56 = 225 \]

Now, substituting into the quadratic formula: \[ x = \frac{13 \pm \sqrt{225}}{4} = \frac{13 \pm 15}{4} \]

Calculating the two possible values for \(x\):

1. \(x_1 = \frac{13 + 15}{4} = \frac{28}{4} = 7\)
2. \(x_2 = \frac{13 - 15}{4} = \frac{-2}{4} = -\frac{1}{2}\)

Final Answer