5)Solve 2x2 +6x = 4 by completing the square.
6)Solve 3x2 - 6x- 24 = 0 using the quadratic formula.
7)The area of a circle is 25 in2. What is the radius, including the units? Area of a circle is ¢³r2.
8)The triangle has an area of 14 in2. Determine the lengths of the sides of the triangles. Remember that the area of a triangle is ¨ö bh.
5) first simplify by dividing by 2
x^2 + 3x = 2
x^2 + 3x + 9/4 = 2 + 9/4
(x+3/2)^2 = 17/4
x+3/2 = √17 /2
x = (-3 ± √17)/2
6) If you are studying this topic, this should be routine
7) π r^2 = 25
r^2 = 25/π
r = 5/√π
8) you don't say what kind of triange.
Is it isosceles, right-angled, scalene ..... ?
5) Solve 2x^2 + 6x = 4 by completing the square:
Step 1: Move the constant term to the right side of the equation:
2x^2 + 6x - 4 = 0
Step 2: Divide the entire equation by the coefficient of x^2 to make the leading coefficient equal to 1:
x^2 + 3x - 2 = 0
Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 + 3x + (3/2)^2 = 2 + (3/2)^2
x^2 + 3x + 9/4 = 8/4 + 9/4
x^2 + 3x + 9/4 = 17/4
Step 4: Factor the left side of the equation:
(x + 3/2)^2 = 17/4
Step 5: Take the square root of both sides of the equation:
x + 3/2 = ±√(17/4)
Step 6: Solve for x by subtracting 3/2 from both sides:
x = -3/2 ± √(17/4)
6) Solve 3x^2 - 6x - 24 = 0 using the quadratic formula:
Step 1: Identify the coefficients of the quadratic equation:
a = 3, b = -6, c = -24
Step 2: Substitute the values of a, b, and c into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Step 3: Calculate the discriminant (b^2 - 4ac):
Discriminant = (-6)^2 - 4(3)(-24) = 36 + 288 = 324
Step 4: Determine the square root of the discriminant:
√324 = 18
Step 5: Plug the values into the quadratic formula and simplify:
x = (-(-6) ± 18) / (2(3))
x = (6 ± 18) / 6
Step 6: Simplify the expression:
x = (6 + 18) / 6 or x = (6 - 18) / 6
x = 24 / 6 or x = -12 / 6
x = 4 or x = -2
Therefore, the solutions to the equation 3x^2 - 6x - 24 = 0 are x = 4 and x = -2.
7) The area of a circle is 25 in^2. What is the radius, including the units?
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
Given that the area A is 25 in^2, we can substitute it into the formula:
25 = πr^2
Divide both sides of the equation by π:
25/π = r^2
Take the square root of both sides of the equation:
r = √(25/π)
Simplify the equation by evaluating the square root and dividing:
r = √25 / √π
r = 5 / √π
Therefore, the radius of the circle is 5/√π inches.
8) The triangle has an area of 14 in^2. Determine the lengths of the sides of the triangle. Remember that the area of a triangle is 1/2 times the base times the height.
Given that the area A is 14 in^2, we can substitute it into the formula for the area of a triangle to find the product of the base and the height:
14 = (1/2)bh
To find the lengths of the sides of the triangle, we need additional information, such as the shape of the triangle (e.g., equilateral, isosceles, or scalene) or another given parameter (e.g., the lengths of two sides or an angle). Without this information, it is not possible to determine the lengths of the sides of the triangle.
5) To solve 2x^2 + 6x = 4 by completing the square, we need to follow these steps:
Step 1: Move the constant term to the other side of the equation:
2x^2 + 6x - 4 = 0
Step 2: Divide the entire equation by the leading coefficient (2 in this case):
x^2 + 3x - 2 = 0
Step 3: Take half of the coefficient of x (3) and square it:
(3/2)^2 = 9/4
Step 4: Add the result from step 3 to both sides of the equation:
x^2 + 3x - 2 + 9/4 = 9/4
x^2 + 3x + 1/4 = 9/4
Step 5: Rewrite the left side of the equation as a perfect square trinomial:
(x + 3/2)^2 = 9/4
Step 6: Take the square root of both sides:
x + 3/2 = ±√(9/4)
x + 3/2 = ±(3/2)
Step 7: Solve for x:
Case 1:
x + 3/2 = 3/2
x = 3/2 - 3/2
x = 0
Case 2:
x + 3/2 = -3/2
x = -3/2 - 3/2
x = -3
Therefore, the solutions are x = 0 and x = -3.
6) To solve 3x^2 - 6x - 24 = 0 using the quadratic formula, we need to follow these steps:
Step 1: Identify the coefficients of the quadratic equation:
a = 3, b = -6, c = -24
Step 2: Apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Step 3: Substitute the values into the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4(3)(-24))) / (2(3))
Step 4: Simplify the expression inside the square root:
x = (6 ± √(36 + 288)) / 6
Step 5: Complete the calculations under the square root:
x = (6 ± √(324)) / 6
x = (6 ± 18) / 6
Step 6: Simplify the expression:
Case 1:
x = (6 + 18) / 6
x = 24 / 6
x = 4
Case 2:
x = (6 - 18) / 6
x = -12 / 6
x = -2
Therefore, the solutions are x = 4 and x = -2.
7) To find the radius of a circle when given the area, you can use the formula A = πr^2.
In this case, we are given that the area (A) of the circle is 25 in^2.
Step 1: Substitute the area into the formula:
25 = πr^2
Step 2: Divide both sides by π to isolate r^2:
r^2 = 25/π
Step 3: Take the square root of both sides to solve for r:
r = √(25/π)
Step 4: Simplify the expression:
r = 5/√π
So, the radius of the circle is 5/√π in units.
8) To determine the lengths of the sides of a triangle when given the area, you can use the formula A = 1/2 * base * height.
In this case, we are given that the area (A) of the triangle is 14 in^2.
Step 1: Substitute the area into the formula:
14 = 1/2 * base * height
Step 2: Rearrange the formula to solve for the base:
base = 2 * (14 / height)
Step 3: Choose a height value that satisfies the condition of the problem. For example, if we assume the height is 7 inches, then plug it into the formula:
base = 2 * (14 / 7)
base = 2 * 2
base = 4
Therefore, when the height is 7 inches, the base is 4 inches.