Problem Solving With Quadratic Equations

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Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.

When we solved linear equations, if an equation had too many fractions we ‘cleared the fractions’ by multiplying both sides of the equation by the LCD.

This gave us an equivalent equation—without fractions—to solve.

We can use the same strategy with quadratic equations. We know from the Zero Products Principle that this equation has only one solution: .

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If you're behind a web filter, please make sure that the domains *.and *.are unblocked.Now, we will go through the steps of completing the square in general to solve a quadratic equation for .It may be helpful to look at one of the examples at the end of the last section where we solved an equation of the form as you read through the algebraic steps below, so you see them with numbers as well as ‘in general.’ This last equation is the Quadratic Formula.That can happen, too, when using the Quadratic Formula.If we get a radical as a solution, the final answer must have the radical in its simplified form.This quantity is called the When the discriminant is positive the quadratic equation has two solutions.When the discriminant is zero the quadratic equation has one solution.We will see in the next example how using the Quadratic Formula to solve an equation with a perfect square also gives just one solution.When we solved the quadratic equations in the previous examples, sometimes we got two solutions, sometimes one solution, sometimes no real solutions.Since I'll be cutting equal-sized squares out of all of the corners, and since the box will have a square bottom, I know I'll be starting with a square piece of cardboard.In this last exercise above, you should notice that each solution method gave the same final answer for the cardboard's width.

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