I can't remember the abbreviation for decimeter!
Abbreviation: dm
thanks
hey....
the abb is dm..
1. Factor 4m²+40m+100
2. The area of a circle is given by the equation A = ð(25x²+10x+1). Find an expression for the radius.
3. The bed of a pond can be modelled by 30 y= x²-15x+14 where x and y are measured by meters and the x-axis matches the water level of the pond. What is the width of the pond?
4. Find the term to create a perfect square trinomial: x² + 22x
5. Complete the square: r² - 4r - 7 = 0
6. The length of a rectangular swimming pool s x + 30, where x = width. It's area is 2800 square feet. Find the length and the width.
Here are the explanations of how to find the answers to your questions:
1. To factor the expression 4m²+40m+100, you can use the method of factoring by grouping or the quadratic formula. Factoring by grouping involves splitting the middle term of the quadratic into two terms in such a way that they can be factored separately. In this case, you can rewrite the expression as (4m²+20m)+(20m+100). Now factor out the common factors from each group, which gives you 4m(m+5)+20(m+5). Finally, you can factor out the common factor (m+5) to get the factored form of the expression: (m+5)(4m+20) or (m+5)(4(m+5)).
2. The equation for the area of a circle is A = πr², where A represents the area and r represents the radius. In this case, the equation A = π(25x²+10x+1) represents the area of the circle. To find an expression for the radius, you need to rearrange the equation in terms of r. Divide both sides of the equation by π, then take the square root of both sides to eliminate the squared term, which gives you r = √(25x²+10x+1).
3. The equation 30 y = x²-15x+14 represents the model for the bed of a pond, where x and y are measured in meters and the x-axis corresponds to the water level of the pond. To find the width of the pond, you need to determine the x-values at which the equation equals zero. Set y = 0 and solve for x. In this case, you have 30(0) = x²-15x+14. Simplifying gives you x²-15x+14 = 0. Factor the quadratic equation to get (x-1)(x-14) = 0. Therefore, the x-values are x = 1 and x = 14. The width of the pond would be the difference between these two x-values, which is 14-1 = 13 meters.
4. To create a perfect square trinomial, you need to find the term that, when added to the given expression x²+22x, completes the square. Take half of the coefficient of the x-term and square it. Half of 22 is 11, and 11 squared is 121. Therefore, the term needed to complete the square is 121. Adding this term to the expression x²+22x gives you x²+22x+121, which is a perfect square trinomial: (x+11)².
5. To complete the square for the quadratic equation r²-4r-7=0, you need to rewrite it in the form (r-h)²+k=0, where h and k represent constants. First, move the constant term to the other side of the equation to isolate the squared and linear terms, which gives you r²-4r=7. Next, take half of the coefficient of the linear term (-4), square it, and add it to both sides of the equation. Half of -4 is -2, and (-2)² is 4. Adding 4 to both sides gives you r²-4r+4=7+4. Simplifying further gives you (r-2)²=11. Therefore, the equation (r-2)²=11 represents the completed square, where h=-2 and k=11.
6. The length of a rectangular swimming pool is represented by s, and the width is represented by x. The area of the pool is given as 2800 square feet. The formula for the area of a rectangle is A = length * width. In this case, you have the equation 2800 = s * x + 30. To find the length and the width, you need to solve this equation for s and x. Since the problem states that the length is s = x + 30, substitute this value into the equation to get 2800 = (x + 30) * x. Simplifying further gives you the quadratic equation x² + 30x - 2800 = 0. Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of x. Once you have the value of x, substitute it into the equation s = x + 30 to find the length.