Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary. List the answer from smallest to largest number.

Question

(x + 27)2−6 = −3

Here is a number line to help you remember the order of numbers.

(1 point)
The first solution is x=

The second solution is x=

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Answer 1

To solve the quadratic equation $(x+27{)}^2-6=-3$, we will first isolate the squared term.

1. Add 6 to both sides:

$$(x+27{)}^2-6+6=-3+6$$

$$(x+27{)}^2=3$$

2. Take the square root of both sides:

$$x+27=\pm \sqrt{3}$$

3. Isolate $x$:

For the positive square root:

$$x=-27+\sqrt{3}$$

For the negative square root:

$$x=-27-\sqrt{3}$$

4. Calculate the numerical values of the solutions:

Using the approximation $\sqrt{3}\approx 1.732$:

For $x=-27+\sqrt{3}$:

$$x\approx -27+1.732\approx -25.268$$

For $x=-27-\sqrt{3}$:

$$x\approx -27-1.732\approx -28.732$$

5. Rounding to the nearest hundredth:

• The first solution (the smaller one) rounded is: $$x\approx -28.73$$

• The second solution (the larger one) rounded is: $$x\approx -25.27$$

6. Listing the answers from smallest to largest:

The first solution is $x=-28.73$

The second solution is $x=-25.27$

Final answers: The first solution is $x=-28.73$

The second solution is $x=-25.27$