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2 Tutorials that teach The Quadratic Formula
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The Quadratic Formula

The Quadratic Formula

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In this lesson, students will learn how to use the quadratic formula to solve quadratic equations with rational solutions.

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Tutorial
This tutorial covers the quadratic formula, through the definition and discussion of:
  1. Quadratic Equations
  2. The Quadratic Formula
  3. Solving Quadratic Equations with the Quadratic Formula


1. Quadratic Equations

Quadratic equations can be written in the form:

a x squared plus b x plus c equals 0

Factoring or variable isolation may be used to solve some, but not all, quadratic equations. However, the quadratic formula can be used to find solutions to all quadratic equations, even when factoring or variable isolation is difficult or impossible. Therefore, sometimes it’s necessary to use the quadratic formula to find solutions to a quadratic equation.

2. The Quadratic Formula

The quadratic formula states that for an equation:

a x squared plus b x plus c equals 0

the solution(s) are:

x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

The values for a, b, and c in the formula come from the values of a, b, and c in the quadratic equation you want to solve. Notice that the equation must be set equal to 0 to get correct values for a, b, and c. When you use the quadratic formula, you often need to simplify square roots.

Note the plus/minus symbol, which indicates that you must evaluate the expression twice—once using addition and once using subtraction—which will lead to two solutions.
KEY FORMULA
x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction


3. Solving Quadratic Equations with the Quadratic Formula

The following example illustrates a real-life application of solving a quadratic equation with a quadratic formula.

IN CONTEXT

Suppose Jason is competing in a shot-put event. How can you determine when the shot hit the ground? The height of the shot is modeled by the equation below, in which x is the time in seconds after the toss, and h of x is the vertical height of the shot.

h left parenthesis x right parenthesis equals equals 16 x squared plus 20 x plus 6

Now, you want to find when the shot hit the ground, which means you want to know when the height is 0. Therefore, the equation you want to solve is the following:

negative 16 x squared plus 20 x plus 6 equals 0

You can solve this equation using the quadratic formula. Comparing your equation to the form for a standard quadratic equation, you can see that:
  • a=-16
  • b=20
  • c=6
File:1349-quadform1.PNG

Therefore, substituting these values into the formula gives you the expression:

x equals fraction numerator negative 20 plus-or-minus square root of left parenthesis 20 right parenthesis squared minus 4 left parenthesis negative 16 right parenthesis left parenthesis 6 right parenthesis end root over denominator 2 left parenthesis negative 16 right parenthesis end fraction

Simplifying the denominator is an easy first step.

x equals fraction numerator negative 20 plus-or-minus square root of 20 squared minus 4 left parenthesis negative 16 right parenthesis left parenthesis 6 right parenthesis end root over denominator negative 32 end fraction

When you simplify the numerator, however, it’s more complicated because it involves a plus/minus symbol, square roots, and other operations. You’ll start to simplify underneath the square root and begin with your exponent, 20^2, then move on to the multiplication, then finally subtraction. Note that 784 is a perfect square, so you’ll finish by taking the square root.

File:1350-quadform2.PNG

Now your expression becomes:

fraction numerator negative 20 plus-or-minus 28 over denominator negative 32 end fraction

You will need to separate your solution into two fractions.

fraction numerator negative 20 over denominator negative 32 end fraction plus-or-minus fraction numerator 28 over denominator negative 32 end fraction

It’s important to remember, though, that the plus and minus symbols gives you two solutions.

x equals fraction numerator negative 20 over denominator negative 32 end fraction plus fraction numerator 28 over denominator negative 32 end fraction space space space space space space space space space space x equals fraction numerator negative 20 over denominator negative 32 end fraction minus fraction numerator 28 over denominator negative 32 end fraction

Simplifying these two fractions provides:

x equals 20 over 32 minus 28 over 32 space space space space space space space space space space x equals 20 over 32 plus 28 over 32

Add and subtract the fractions accordingly, then divide to provide your two solutions to this quadratic equation.

x equals fraction numerator negative 8 over denominator 32 end fraction equals negative 0.25 space space space space space space space space space space x equals 48 over 32 equals 1.5

Going back to your original problem, you want to find the time it takes to reach the ground. Your solution of x equals -0.25 can be disregarded because it doesn’t make sense in the context of the problem. Therefore, it takes 1.5 seconds for the shot to hit the ground.

Today you learned that factoring or variable isolation may be used to solve some, but not all, quadratic equations. However, the quadratic formula can be used to find solutions to all quadratic equations, even when factoring or variable isolation is difficult or impossible. It’s important to note that before solving quadratic equations with the quadratic formula, the equation must be equal to 0 to determine the correct values of a, b, and c to use in the formula.

Source: This work is adapted from Sophia author Colleen Atakpu.

TERMS TO KNOW
  • KEY FORMULA

    x= [=-b± sqrt(b^2-4ac)]/2a