1. What is the solution of n² – 49 = 0? (1 point)
–7
7
±7
no solution
2. What is the solution of x² + 64 = 0? (1 point)
–5
8
±8
no solution
3. What is the side length of a square with an area of 144x²? (1 point)
12
12x
±12x
no solution
4. What is the value of b in the triangle shown below?
(Image: A right triangle is shown. The left leg equals 3 b. The lower leg equal b. The area of the triangle equals 24 square inches.) (1 point)
–4 in
4 in
±4 in
no solution
5. What is the value of z so that –9 and 9 are both solutions of x² + z = 103?
–22
3
22
184
answer
1. ±7
2. no solution
3. ±12x
4. 4 in
5. 22
for #3, -12 cannot be a side length
1. To solve the equation n² – 49 = 0, we can start by adding 49 to both sides of the equation:
n² = 49
2. Now, we can take the square root of both sides of the equation to solve for n:
√(n²) = ±√49
n = ±7
Therefore, the solutions to the equation are ±7.
2. To solve the equation x² + 64 = 0, we can start by subtracting 64 from both sides of the equation:
x² = -64
3. Now, we can take the square root of both sides of the equation to solve for x:
√(x²) = ±√(-64)
Since √(-64) is not a real number, there is no real solution to the equation. Therefore, the answer is no solution.
3. The area of a square is equal to the side length squared, so to find the side length, we can take the square root of the area.
√(144x²) = ±12x
Therefore, the side length of the square is ±12x.
4. To find the value of b in the triangle, we can use the formula for the area of a triangle:
Area = 1/2 * base * height
In this case, the area is given as 24 square inches, and the base is given as b. The height is given as 3b.
24 = 1/2 * b * 3b
Simplifying the equation, we get:
24 = 1.5b²
Dividing both sides of the equation by 1.5, we get:
16 = b²
Taking the square root of both sides of the equation, we get:
√(16) = ±4
Therefore, the value of b in the triangle is ±4 inches.
5. We are given that –9 and 9 are both solutions of the equation x² + z = 103. This means that when we substitute –9 and 9 into the equation, the equation will be true.
Let's substitute –9 into the equation:
(-9)² + z = 103
Simplifying, we get:
81 + z = 103
Subtracting 81 from both sides of the equation, we get:
z = 103 - 81
z = 22
Therefore, when z = 22, the equation is true for –9.
Let's substitute 9 into the equation:
9² + z = 103
Simplifying, we get:
81 + z = 103
Subtracting 81 from both sides of the equation, we get:
z = 103 - 81
z = 22
Therefore, when z = 22, the equation is true for 9.
Therefore, the value of z so that –9 and 9 are both solutions of x² + z = 103 is 22.
1. To find the solution of n² - 49 = 0, we can factor the equation using the difference of squares property. The difference of squares property states that a² - b² = (a + b)(a - b).
In this case, n² - 49 can be written as (n + 7)(n - 7).
Setting each factor equal to zero, we get:
n + 7 = 0 or n - 7 = 0
Solving these equations, we find:
n = -7 or n = 7
Therefore, the solutions of n² - 49 = 0 are ±7.
2. To find the solution of x² + 64 = 0, we can subtract 64 from both sides of the equation:
x² = -64
To solve for x, we can take the square root of both sides:
x = ±√(-64)
There is no real number solution for the square root of a negative number. Therefore, the given equation x² + 64 = 0 has no solution.
3. The area of a square is calculated by squaring the length of its side. In this case, the area of the square is given as 144x².
To find the side length of the square, we need to take the square root of the area:
√(144x²) = √(12² * x²) = 12x
Therefore, the side length of the square is 12x.
4. To find the value of b in the given triangle, we can use the formula for the area of a right triangle:
Area = (base * height) / 2
In this case, the area of the triangle is given as 24 square inches. The base of the triangle is equal to b, and the height is equal to 3b.
Plugging in these values into the formula, we get:
24 = (b * 3b) / 2
Multiplying both sides of the equation by 2, we get:
48 = 3b²
Dividing both sides of the equation by 3, we get:
16 = b²
Taking the square root of both sides of the equation, we get:
b = ±4
Therefore, the value of b in the triangle is ±4 inches.
5. To find the value of z that satisfies the equation x² + z = 103 for both x = -9 and x = 9, we can substitute these values into the equation and solve for z.
When x = -9:
(-9)² + z = 103
81 + z = 103
z = 103 - 81
z = 22
When x = 9:
(9)² + z = 103
81 + z = 103
z = 103 - 81
z = 22
Therefore, the value of z that satisfies the equation for both x = -9 and x = 9 is 22.