By HOLT MCDOUGAL
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Extra resources for Math Resource. Algebra
Sample text
A) 8x = 12ix + 13 b) 3x2 + 4ix = 12 c) x2 = i Solutions: a) This equation is a linear equation so we need only to solve for x. 8x – 12ix = 13 ← add the additive inverse of 12ix to each side x(8 – 12i) = 13 ← factor x to try to isolate the variable 13 ← multiply by the mult. inverse of 8 − 12i x= 8 − 12i 13 8 + 12i ⋅ ← multiply by a clever form of 1 to simplify x= 8 − 12i 8 + 12i 104 + 156 i ← perform the binomial multiplication x= 64 − 144i 2 104 + 156 i ← i 2 = −1 x= 208 2 + 3i ← simplify the fraction x= 4 1 3 x= + i ← put the answer in a + bi form 2 4 b) This is a quadratic equation, and we will need the quadratic formula to solve.
Quite intuitively, if three quantities multiply together to give zero as their product, then at least one of those three numbers must be zero. It may not be apparent at first how this can apply to the solution of quadratic equations, but a few examples will drive the point home. Examples: Solve the following equations by factoring. a) x2 – x + 12 = 0 b) 6x2 – x – 12 = 0 c) x2 = 81 Solutions: a) This is a simple quadratic equation. After a bit of searching, we can factor the expression on the left to create an equivalent equation.
In this case, gut instinct leads us to x = 11 and x = -11. General Formula for a Squared Variable If we are given that x2 = c, where c > 0, then we solve for x by saying that x = ± c . Example: Use the general formula for a squared variable to solve (x + 2)2 = 50. Solution: (x + 2)2 = 50 x + 2 = ± 50 ← general formula for a squared variable x+2=± 5 2 ← simplify the square root x = -2 ± 5 2 ← add the additive inverse of 2 to each side x = -2 + 5 2 or x = -2 – 5 2 We can use this idea to our advantage in an algebraic technique known as completing the square.
Math Resource. Algebra by HOLT MCDOUGAL
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