Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-factor property is then used to find solutions. See
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Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See
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Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See
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Completing the square is a method of solving quadratic equations when the equation cannot be factored. See
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A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See
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The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See
[link].
The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See
[link].
Section exercises
Verbal
How do we recognize when an equation is quadratic?
It is a second-degree equation (the highest variable exponent is 2).
When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form
we may graph the equation
and have no zeroes (
x -intercepts).
When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?
We want to take advantage of the zero property of multiplication in the fact that if
then it must follow that each factor separately offers a solution to the product being zero:
In the quadratic formula, what is the name of the expression under the radical sign
and how does it determine the number of and nature of our solutions?
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
