Which two of the following values are solutions to the quadratic equation below? 𝑥2−3𝑥−3=7x 2 −3x−3=7A

1. Break Down the Problem

To find the solutions to the quadratic equation $x^2 - 3x - 3 = 7$, we first need to rearrange it into standard form by moving all the terms to one side of the equation.

2. Relevant Concepts

The standard form of a quadratic equation is given by: $ax^2 + bx + c = 0$ We will solve for $x$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3. Analysis and Detail

1. Rearrange the equation: $x^2 - 3x - 3 - 7 = 0 \implies x^2 - 3x - 10 = 0$ Here, we identify $a = 1$, $b = -3$, and $c = -10$.

2. Calculate the discriminant: $D = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-10) = 9 + 40 = 49$

3. Apply the quadratic formula: $x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 1} = \frac{3 \pm 7}{2}$

   • For $x_1$: $x_1 = \frac{3 + 7}{2} = \frac{10}{2} = 5$
   • For $x_2$: $x_2 = \frac{3 - 7}{2} = \frac{-4}{2} = -2$

4. Verify and Summarize

The solutions to the equation $x^2 - 3x - 10 = 0$ are $x = 5$ and $x = -2$. These values have been verified through the calculations.

Final Answer

The two solutions to the quadratic equation $x^2 - 3x - 3 = 7$ are $x = 5$ and $x = -2$.