2.5 Quadratic equations  (Page 6/14)

Finding the length of the missing side of a right triangle

Find the length of the missing side of the right triangle in [link] .

As we have measurements for side b and the hypotenuse, the missing side is a.

$$\begin{array}{ccc}\hfill a^2+b^2& =& c^2\hfill \\ a^2+{(4)}^2\hfill & =& {(12)}^2\hfill \\ \hfill a^2+16& =& 144\hfill \\ \hfill a^2& =& 128\hfill \\ \hfill a& =& \sqrt{128}\hfill \\ & =& 8\sqrt{2}\hfill \end{array}$$

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Use the Pythagorean Theorem to solve the right triangle problem: Leg a measures 4 units, leg b measures 3 units. Find the length of the hypotenuse.

$5$ units

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Key equations

Key concepts

• 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 [link] , [link] , and [link] .
• Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See [link] and [link] .
• 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 [link] and [link] .
• Completing the square is a method of solving quadratic equations when the equation cannot be factored. See [link] .
• A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See [link] .
• 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] .

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