Solve by (a) Completing the square (b) Using the quadratic formula 2 x^2-32 x+7=0

Name: Problem 36, Chapter 9: Algebra Form and Function
Uploaded: 2022-04-07T16:21:25-08:00
Duration: 3 min 34 s
Channel: Alison Rodriguez
Description: Solve by (a) Completing the square (b) Using the quadratic formula 2 x^2-32 x+7=0

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

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two typically expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ? 0. They represent parabolic graphs and arise in various contexts including physics, engineering, and economics, making their solutions fundamental in algebra.

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the quadratic expression in the form of a perfect square trinomial plus or minus a constant. This technique involves manipulating the equation algebraically to isolate the squared term, which then allows one to solve for the variable by taking square roots.

Quadratic Formula

The quadratic formula, derived from the process of completing the square on the general quadratic equation, provides a direct solution for the roots of any quadratic equation. It is given by x = (-b ± ?(b² - 4ac)) / (2a) and enables the determination of the roots by substituting the coefficients of the equation, making it a powerful and universally applicable tool in algebra.

Transcript

00:01 To begin this problem by completing the square, i'm going to move the 7 to the right -hand side.

00:06 So what i would have over here now is negative 7.

00:09 Now from here, we're going to go ahead and take everything and divide it through by 2.

00:14 Because when we do this process, we need to make sure that a is 1.

00:18 So it gets us x squared minus 16x.

00:21 And over here, we're just going to stay as negative 7 over 2.

00:24 Our next steps now is to do the b over 2 squared.

00:27 So i have negative 16 over 2 and square it.

00:30 So this becomes 64.

00:31 So i'm going to do plus 64 on this side and also plus 64 on that side.

00:36 So now on the right hand side it's factorable.

00:39 So this now becomes x minus 8 to the second power, so x minus 8 squared.

00:43 For this one, we're going to change this essentially into 128 over 2 in order for us to combine that together.

00:50 So really we're doing 128 minus 7.

00:54 And what that will leave us with is 121 over 2.

00:58 All right.

00:59 So now i'm ready to take the square rootable.

01:01 Sides.

01:03 And so i have x minus eight on the left equals plus or minus 11 over the square root of two.

01:08 So i took the square to 121 because it was a perfect square, but the two i could not.

01:13 So what we could do, though, is rationalize.

01:15 You should always be rationalizing here.

01:17 So i'm going to do times a square root of two at the top and the bottom.

01:21 So this will get you x minus 8 equals plus or minus 11 square root of two.

01:27 Let me make that a little bit neater here with the plus or minus, looks like a number symbol, all over two.

01:34 So now i'm going to put this over onto the right -hand side, moving that 8 over...

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