How to Solve Quadratic Equations by Factoring (NancyPi) @NancyPi

NancyPi | How to Solve Quadratic Equations by Factoring (NancyPi) @NancyPi | Uploaded May 2018 | Updated October 2024, 3 hours ago.
MIT grad shows how to solve any quadratic equation by factoring. To skip to the shortcut trick, go to time 6:11. Nancy formerly of MathBFF explains the steps.

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The shortcut trick ("The Magic X") helps you factor any tough quadratic that doesn't begin with x^2 but instead begins w/ 2x^2 or 3x^2, or 4x^2, etc, so you can then solve.

1) IF QUADRATIC STARTS WITH X^2: It's faster to use the normal method for factoring in this case: trial and error. Ex: x^2 + 4x - 12. Find 2 numbers that multiply to the last number, -12, AND that add to the second coefficient, positive 4. First make a list of all pairs of #s that multiply to -12. Then check which pair also adds to 4. Write factors & solve by setting each = 0. Solve for x.

2) IF QUADRATIC STARTS WITH 2X^2 OR 3X^2 OR 4X^2, ETC:
A) First check if leading coefficient (2 or 3 or 4, etc) is an overall constant you can factor out of every term. If it is, factor it out first, then use Method #1 above to factor the X^2 expression that's left. Set factors = 0 & solve.
B) If a constant can't be factored out evenly from every term, it'll be faster & easier to use shortcut "magic X" method instead of Method #1. See it explained at time 6:11 in video. Set factors = 0 & solve.

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$Logarithms... How? (NancyPi) MIT grad introduces logs and shows how to evaluate them. To skip ahead: 1) For how to understand and evaluate BASIC LOGS, skip to time 0:52. 2) For how to evaluate weirder logs, including the log of 0, 1, a FRACTION, or a NEGATIVE number, skip to time 6:44. 3) For NATURAL LOGS (LN X), skip to time 11:17. 4) For even weirder logs, including SOLVING for X and using the CHANGE-OF-BASE formula, skip to time 14:56. Nancy formerly of MathBFF explains the steps. Follow Nancy on Instagram: https://instagram.com/nancypi Twitter: https://twitter.com/nancypi 1) BASIC LOGS: you can read log notation as log, base 3, of 9 equals X. The small (subscript) number is called the base. You can always evaluate a log expression by rearranging it into something called exponential form. Every log expression is connected to an exponential expression. In this example, the log is connected to the exponential form 3 to the X power equals 9. This means, 3 raised to what power gives you 9? Since 3 raised to the power of 2 equals 9, the answer for X is 2. This is also the answer for the value of the log expression. The log is always equal to the power (or exponent) in the exponential version, and in this case it equals 2. If you want, you can find the log value in your head just by asking yourself what power you need in order to turn the base number into the middle number (argument number). Note: if there is no base number in the log expression (no little subscript number), then the base is 10, since 10 is the default base. 2) WEIRDER LOGS (log of 0, 1, a negative number, or a fraction): you can use the same steps to rearrange log expressions that have a fraction, negative number, 0, or 1 in them. You can still rearrange them to be in exponential form just like you can with the basic logs from earlier. The log of 1 will always be 0, since 0 is the only power that can turn a base into 1. The log of 0 will always be undefined, since no power can turn a base into 0. The log of a negative number is undefined in the real number system, since no real power can turn a positive base into a negative number. 3) NATURAL LOGS (ln x): the natural log is just a special type of log where the base is e (the special math constant e, which is approximately 2.718 if you plug it into your calculator). You can use the same steps for rearranging the log expression into exponential form. Just remember that ln x means log, base e. 4) EVEN WEIRDER LOGS (solving for X, change-of-base formula): even if there is an X variable in the log part of an equation, you can still rearrange the equation into exponential form. This will let you solve for X. Sometimes you might need to use the change-of-base formula to evaluate a log expression. If there is no whole number power you know that works, it may actually be a decimal power that you can find by using the change-of-base formula. For example, you can re-write log, base 2, of 7 as (log 7)/(log 2) and use your calculator to find the decimal number if you need it. For more of my math videos, check out: http://nancypi.com$

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$Basic Trigonometry: Sin Cos Tan (NancyPi) MIT grad shows how to find sin, cos, and tan using SohCahToa as well as the csc, sec, and cot trig functions. To skip ahead: 1) For how to find the adjacent, opposite, and hypotenuse sides of the right triangle, skip to 1:13. 2) For how to find the value of SIN of theta using SOH CAH TOA, skip to 3:16. 3) For how to find COS, skip to 4:15. 4) For how to find TAN, skip to 4:56. 5) For how to find CSC, SEC, AND COT from the sin, cos, and tan functions, skip to 5:56. 6) For harder examples of how to label the adj, opp, and hyp sides if the TRIANGLE IS ROTATED on its side or upside down, skip to 8:32. Nancy formerly of MathBFF explains trigonometry basics. For MORE TRIG: how to convert between radians and degrees, jump to: https://youtu.be/l6hSY2Pcch0 Follow me on Instagram! https://instagram.com/nancypi Follow me on Twitter: https://twitter.com/nancypi SIN COS TAN: If you have a right triangle with side lengths given and an angle (theta), you can easily find the sine, cosine, and tangent trigonometric functions. Remember that the longest side of the triangle is the HYPOTENUSE, the side directly opposite theta is the OPPOSITE, and the other side next to theta is the ADJACENT side. Once youve identified the hypotenuse (H), adjacent (A), and opposite sides (O), you can find Sin, Cos, and Tan using these trigonometric ratios: Sin of theta = Opposite over Hypotenuse (SOH) Cos of theta = Adjacent over Hypotenuse (CAH) Tan of theta = Opposite over Adjacent (TOA) The mnemonic SOH CAH TOA can help you remember which sides to use for the sine, cosine, and tangent trig ratios. For instance, SOH stands for Sin = Opposite over Hypotenuse CSC SEC COT: If you need to find one of the other three trig functions (cosecant, secant, or cotangent), the easiest way to find them is to first find Sin, Cos, or Tan and then take the reciprocal (or flip) the fraction, since: Csc = 1 / Sin Sec = 1 / Cos Cot = 1 / Tan For example, if you find that Cos of an angle is equal to 3/5, the Sec of that angle would be equal to 5/3. For more of my mathematics videos, check out: http://nancypi.com$

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$Basic Integration... How? (NancyPi) MIT grad shows how to find antiderivatives, or indefinite integrals, using basic integration rules. To skip ahead: 1) For how to integrate a polynomial with the POWER RULE, skip to 1:35. 2) For how to integrate NEGATIVE POWERS of x, FRACTIONAL POWERS of x, and RADICALS/ROOTS, skip to 6:12. 3) For how to integrate x^(-1), or 1/x, with the LOG RULE, skip to 7:23. 4) For examples where you use more algebra to rewrite before integrating, to SEPARATE the numerator of a FRACTION, or EXPAND a PRODUCT in order to use the Power Rule, skip to 10:00. 5) For basic TRIG and EXPONENTIAL examples that use rules from the Table of Integrals, as well as trig identities, skip to 11:36. Nancy formerly of MathBFF explains the steps. For my video on how to integrate using U-SUBSTITUTION, jump to: https://youtu.be/8B31SAk1nD8 For my video on how to do INTEGRATION BY PARTS, jump to: https://youtu.be/KKg88oSUv0o Follow Nancy on Instagram: https://instagram.com/nancypi Twitter: https://twitter.com/nancypi 1) POWER RULE: If youre integrating a polynomial, or just a power of x, you can use the Power Rule on each x-term in the polynomial. The Power Rule says that for a term thats just a power of x, such as x^3, you can integrate by raising the power by 1 AND dividing by that number (the new number you got by increasing the power by 1). For example, the integral of x^3 would be (x^4)/4. If there was a constant multiplied in front of the x-power, you can keep the constant and integrate the rest of the term (Constant Multiple Rule). For example, the integral of 6x^2 would be 6 times (x^3)/3, which simplifies to 2x^3. You can repeat these steps to integrate every term in the polynomial and string them together for the full integral. You can also keep the addition and subtraction between the terms (because of the Sum and Difference Rules). NOTE: The anti-derivative of a constant (just a number) is always that number times x (the Constant Rule), so the integral of 1 is 1*x, or x. At the end, its very important to remember to add a constant of integration, to include a + c at the end of your answer, when it is an indefinite integral (integral with no limits). ANOTHER NOTE: The Power Rule only works when the power is not -1. 2) NEGATIVE & FRACTIONAL POWERS, and RADICALS: For negative powers, you can still use the Power Rule, as long as the power is not -1. For fractional powers, you can also use the Power Rule. When you increase a fractional power by 1, you will have to simplify the power (before integrating) by getting a common denominator. For roots, like the square root of x, its best to rewrite the radical as a fractional power, and then use the Power Rule. 3) IF THE X POWER IS -1 in the integrand, either written as x^(-1) or 1/x, you have to use a special rule, the LOG RULE, that you can find on a Table of Integrals. It says that the anti-derivative of x^(-1), or (1/x) is the natural log of the absolute value of x, plus c. 4) MORE ALGEBRA: Sometimes you may need to do a bit more algebra before integrating, so that the integrand is in a form that fits a basic integration rule, like the Power Rule. If youre integrating a rational expression, you can sometimes separate the numerator, or break the fraction into separate fractions, simplify each term, and then integrate with the Power Rule. If you have a product of x expressions, you can multiply it out, or distribute the factors so that you have just a polynomial (and can use the Power Rule). 5) TRIG & EXPONENTIAL: You can find a lot of trigonometric (and exponential) integral rules in the Table of Integrals. If you dont see it in the table, you may need to use a trig identity first, to rewrite the integrand into a form that you can integrate. For more of my math videos, check out: http://nancypi.com$

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