Multiply by 3-digit numbers

Learning Objectives Accurately square any 3-digit number to find the value of r² for a circle's equation. Apply the distributive property to multiply binomials containing 3-digit numbers, such as (x - h)² where h is a 3-digit integer. Convert the standard form equation of a circle, (x - h)² + (y - k)² = r², into its general form, x² + y² + Dx + Ey + F = 0, when h, k, or r are 3-digit numbers. Calculate the coefficients D, E, and F for the general form of a circle's equation using multiplication with 3-digit numbers. Verify the accuracy of large-number calculations when manipulating circle equations. Determine the constant term F in the general form by combining multiple terms derived from squaring 3-digit numbers. Ever wondered how engineers model a massive radio t...

TermDefinitionExample Standard Form of a CircleThe equation (x - h)² + (y - k)² = r², which explicitly defines a circle with a center at the point (h, k) and a radius of length r.The equation (x - 110)² + (y - 50)² = 900 represents a circle centered at (110, 50) with a radius of 30 (since √900 = 30). General Form of a CircleThe equation x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. This form is derived by expanding the standard form.x² + y² - 220x - 100y + 12200 = 0 is the general form of the circle from the previous example. Center (h, k)The central point from which all points on the circle are equidistant. In the standard form, these are the values being subtracted from x and y.For the circle (x - 125)² + (y + 200)² = 1, the center is (125, -200). Radius (r)The fixed dista...

Standard Form to General Form Conversion (x - h)² + (y - k)² = r² expands to x² + y² - 2hx - 2ky + (h² + k² - r²) = 0 Use this to convert from the standard form to the general form. This requires calculating 2h, 2k, h², k², and r², which often involves multiplying by 3-digit numbers. General Form Coefficients D = -2h, E = -2k, F = h² + k² - r² These formulas define the direct relationship between the center (h, k) and radius (r) from the standard form and the coefficients D, E, and F in the general form. Calculating F is where precise multiplication and addition of large numbers are crucial.