Solve by (a) Completing the square (b) Using the quadratic formula 5 x^2+17 x+1=0

Name: Problem 38, Chapter 9: Algebra Form and Function
Uploaded: 2022-04-07T16:29:07-08:00
Duration: 2 min 52 s
Channel: Alison Rodriguez
Description: Solve by (a) Completing the square (b) Using the quadratic formula 5 x^2+17 x+1=0

Solve by (a) Completing the square (b) Using the quadratic...

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Key Concepts

Completing the Square

This method involves rewriting a quadratic expression as a perfect square plus or minus a constant. It is a powerful algebraic technique used not only for solving quadratic equations but also for analyzing conic sections and integrating certain functions. By transforming the quadratic into a perfect square form, one can easily solve for the variable through taking square roots of both sides.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable, typically expressed in the form ax² + bx + c = 0. It is fundamental in algebra and appears in various contexts such as physics, engineering, and economics. Solving quadratic equations involves finding the values of the variable that satisfy the equation.

Quadratic Formula

The quadratic formula provides a direct way to find the solutions of any quadratic equation by substituting the coefficients into the formula x = (-b ± ?(b² - 4ac)) / (2a). It is derived by completing the square on the general quadratic equation and is particularly useful because it encapsulates all possible cases, including real and complex roots, based on the value of the discriminant.

Transcript

00:01 To begin the problem by completing the square, we move the c value over to the right -hand side.

00:05 So it becomes 5x square plus 17x is equal to negative 1.

00:10 We're going to have quite a few fractions here, so take it step by step as we go.

00:14 We're going to divide everything through by 5 over here on the left -hand side, which is going to leave us with a couple of fractions.

00:19 The first thing is, though, is still just x -square, which is required.

00:23 Then we have 17 -5s x.

00:25 We're going to leave this space, and over here we'll have negative 1 over 5.

00:28 Our next step now is to do the b over 2 squared.

00:32 Keep in mind that dividing by 2 is the same thing as multiplying by a half, which might be something easier for us to visualize.

00:38 So this becomes on the bottom 10th squared, which is 100, and the numerator 17 squared, which is 289.

00:50 So i'm going to add 289 over 100 for both of these.

00:55 So plus 289 over 100.

00:59 All right, even though, again, this is really where patterns are going to come into play to remember that this is always the b over 2 here.

01:06 So this will be 17 tenths, 17 tenths squared.

01:12 So the square root of 289 over 100 as well.

01:14 Now over here with that fraction, we want to think about, okay, i have this negative 1 over 5...

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