For the quadratic equation (x−7)^2=11 , complete the table by testing each of the given values to determine whether it is a solution. Identify the two solutions to the quadratic equation. (1 point) The smaller of the two solutions is x= . The larger of the two solutions is x= .

Answer 1

To find the solutions to the quadratic equation $(x-7{)}^2=11$, we first need to solve the equation.

1. Take the square root of both sides: $$x-7=\pm \sqrt{11}$$

2. This gives us two separate equations to solve:

   • $x-7=\sqrt{11}$
   • $x-7=-\sqrt{11}$

3. Solve for $x$ in each case:

   • For $x-7=\sqrt{11}$: $$x=7+\sqrt{11}$$

   • For $x-7=-\sqrt{11}$: $$x=7-\sqrt{11}$$

Now we can identify the two solutions:

• The smaller solution is $x=7-\sqrt{11}$.
• The larger solution is $x=7+\sqrt{11}$.

To approximate these values, we can use $\sqrt{11}\approx 3.3166$:

• Smaller solution: $x\approx 7-3.3166\approx 3.6834$
• Larger solution: $x\approx 7+3.3166\approx 10.3166$

Thus, we summarize the solutions:

• The smaller of the two solutions is $x=7-\sqrt{11}$.
• The larger of the two solutions is $x=7+\sqrt{11}$.

So the final answers are:

• The smaller of the two solutions is $x=7-\sqrt{11}$.
• The larger of the two solutions is $x=7+\sqrt{11}$.