Question

1. solve the following quadratic equation for all values of x in simplest form. 3(x - 8)² = 27 (with some handwritten steps and answer x = 11, -11) 15. solve the following quadratic equation for all values of x in simplest form. (x + 3)² = 15 16. solve the following quadratic equation for all values of x in simplest form. 2(x + 7)² = 2

Explanation:

Problem 14

Step1: Divide both sides by 3

To isolate the squared term, we divide both sides of the equation \(3(x - 8)^2 = 27\) by 3. This gives us \(\frac{3(x - 8)^2}{3}=\frac{27}{3}\), which simplifies to \((x - 8)^2 = 9\).

Step2: Take square root of both sides

Taking the square root of both sides, we have \(\sqrt{(x - 8)^2}=\pm\sqrt{9}\). This simplifies to \(x - 8=\pm3\).

Step3: Solve for x (two cases)

• Case 1: \(x - 8 = 3\). Adding 8 to both sides, we get \(x=3 + 8=11\).
• Case 2: \(x - 8=-3\). Adding 8 to both sides, we get \(x=-3 + 8 = 5\).

Step1: Take square root of both sides

For the equation \((x + 3)^2 = 15\), we take the square root of both sides. This gives us \(\sqrt{(x + 3)^2}=\pm\sqrt{15}\), which simplifies to \(x+3=\pm\sqrt{15}\).

Step2: Solve for x (two cases)

• Case 1: \(x+3=\sqrt{15}\). Subtracting 3 from both sides, we get \(x=\sqrt{15}-3\).
• Case 2: \(x + 3=-\sqrt{15}\). Subtracting 3 from both sides, we get \(x=-\sqrt{15}-3\).

Step1: Divide both sides by 2

To isolate the squared term, we divide both sides of the equation \(2(x + 7)^2 = 2\) by 2. This gives us \(\frac{2(x + 7)^2}{2}=\frac{2}{2}\), which simplifies to \((x + 7)^2 = 1\).

Step2: Take square root of both sides

Taking the square root of both sides, we have \(\sqrt{(x + 7)^2}=\pm\sqrt{1}\). This simplifies to \(x + 7=\pm1\).

Step3: Solve for x (two cases)

• Case 1: \(x + 7 = 1\). Subtracting 7 from both sides, we get \(x=1-7=-6\).
• Case 2: \(x + 7=-1\). Subtracting 7 from both sides, we get \(x=-1 - 7=-8\).

Answer:

\(x = 5\) or \(x = 11\)

Problem 15