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urp:poly
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| Both sides previous revision Previous revision Next revision | Previous revision | ||
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urp:poly [2021-12-19] nerf_herder |
urp:poly [2022-01-07] (current) nerf_herder [Completing the square] |
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| start with quadratic equation: ax² + bx + c = 0 | start with quadratic equation: ax² + bx + c = 0 | ||
| - | **equation of square: a(x+d)² + e = 0,** where d = b/(2a) and e = c - b²/(4a) | + | === Method 1 === |
| + | |||
| + | Rewrite as x² + (b/a)x = -c/a | ||
| + | That becomes (x + b/2a)² = -c/a + (b/2a)² | ||
| + | Call d = b/2a: (x + d)² = d² - c/a | ||
| + | Take square roots: x + d = ±√(d² - c/a), and simplify | ||
| + | (Solving for x gives two answers, call it u and v. | ||
| + | The quadratic equation can then be rewritten as: a(x+u)(x+v) = 0) | ||
| + | |||
| + | === Method 2 === | ||
| + | |||
| + | Just compute the values: **equation of square: a(x+d)² + e = 0,** where d = b/(2a) and e = c - b²/(4a) | ||
| | | ||
| - | steps to get there: divide by a to get x² by itself | + | steps to get there: use method 1 to get (x + d)² = d² - c/a |
| - | x² + (b/a)x + (c/a) = 0 | + | multiply both sides by a: a(x + d)² = ad² - c |
| - | add & subtract ((b/a)/2)², now you have: | + | call e = c - ad², then a(x + d)² + e = 0 |
| - | x² + (b/a)x + ((b/a)/2)² + (c/a) - ((b/a)/2)² = 0 | + | |
| (x + (b/a))² + c/a - ((b/a)/2)² = 0 | (x + (b/a))² + c/a - ((b/a)/2)² = 0 | ||
| \==========/ | \==========/ | ||
| quadratic + remainder = 0 | quadratic + remainder = 0 | ||
| - | -b/a is the vertex (x value) if graphing, y = remainder | + | |
| + | -b/a is the vertex (x value) if graphing, y = remainder | ||
| which leads to **quadratic formula**: x = (-b +- √(b²-4ac)) / 2a | which leads to **quadratic formula**: x = (-b +- √(b²-4ac)) / 2a | ||
| Line 50: | Line 62: | ||
| This has 3 different polynomials for x, and will touch or cross the x axis 3 times | This has 3 different polynomials for x, and will touch or cross the x axis 3 times | ||
| - | ====Rule of signs==== | + | ====Rule of signs & Number of roots==== |
| Roots of a polynomial are locations where y=0. Rule of signs can tell us how many roots exist, and how many have a positive or negative value of x. | Roots of a polynomial are locations where y=0. Rule of signs can tell us how many roots exist, and how many have a positive or negative value of x. | ||
| - | Count the number of times the sign of coefficient of terms changes sign (ignoring any zero coefficients). | + | The highest power is the number of total roots. |
| + | |||
| + | Count the number of times the sign of coefficient of terms changes sign (ignoring any zero coefficients) as you go down the list of coefficients. | ||
| + | |||
| + | If there is no constant term (power of 0), pull out a factor of x, until there is a constant. For example: x³+2x²+5x gets changed to x(x²+2x+5), then use x²+2x+5 in the steps below. (Total number of roots is from original equation: 3 in this case.) | ||
| Example: | Example: | ||
| -3x⁴ + 4x² + x − 2 | -3x⁴ + 4x² + x − 2 | ||
| - | The highest power is 4, so there are 4 roots in total. | + | |
| - | This has two changes in sign, so a maximum of two positive roots (or possibly a pair of complex roots). | + | Positive Roots: This has two changes in sign, so a maximum of two positive roots |
| - | The number of positive roots is reduced by 2 for each pair of complex roots, if they exist. | + | Negative Roots: Substitute -x => only odd powers will change sign. Count the changes in sign again. |
| - | For negative roots, substitute -x => only odd powers will change sign. Count the changes in sign again. | + | |
| In this example, only the x changes: -3(-x)⁴ + 4(-x)² + -x − 2 = -3x⁴ + 4x² -x − 2 | In this example, only the x changes: -3(-x)⁴ + 4(-x)² + -x − 2 = -3x⁴ + 4x² -x − 2 | ||
| still two changes => two negative roots (or 0 if there's complex roots) | still two changes => two negative roots (or 0 if there's complex roots) | ||
| + | The number of positive or negative roots is reduced by 2 for each pair of complex roots, if they exist. | ||
| That means, in this case, there could be: | That means, in this case, there could be: | ||
| * 4 roots: 2 positive roots, 2 negative roots | * 4 roots: 2 positive roots, 2 negative roots | ||
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| When the leading coefficient is 1, the possible rational zeros are the factors of the constant term. | When the leading coefficient is 1, the possible rational zeros are the factors of the constant term. | ||
| + | ------------ | ||
| + | |||
| + | Back to the [[00_start|Start]] page or [[math]] page. | ||
urp/poly.1639949463.txt.gz · Last modified: 2021-12-19 by nerf_herder
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