Solve quadratic equation by completing the square. 2...

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants. The equation represents a parabola when graphed, and solving it means finding the x-values where the parabola crosses the x-axis.

Standard Form

Expressing a quadratic equation in the standard form (ax² + bx + c = 0) is crucial because it provides a structured format that allows for systematic solution methods, such as completing the square or using the quadratic formula.

Completing the Square

Completing the square is an algebraic technique used to solve quadratic equations by rewriting the quadratic expression as a perfect square trinomial plus or minus a constant. This method transforms the equation into a form that can be easily solved by taking square roots.

Perfect Square Trinomial

A perfect square trinomial is an expression that can be factored into the square of a binomial. Recognizing and forming a perfect square is central to the completing the square method, as it simplifies the process of isolating the variable.

Transcript

00:01 This is for problem number 30 of the section 9 .2 in the book introductory algebra for college students.

00:07 First problem, we are given a quadratic equation 2x squared minus x minus 1 equals 0.

00:15 We're told to solve this for x using the complete -square method.

00:20 So our first step for the complete -square method is that we have to have a binomial in the form of x -squared plus bx.

00:28 And right now, we have this constant in the way.

00:31 So we'll add one to both sides to move to an isolate our binomial.

00:35 And we'll be left with 2x squared minus x equals 1.

00:40 Now we need to get rid of this coefficient 2 in front of our x squared.

00:45 So we'll divide everything by 2.

00:47 And we'll get x squared minus 1 1⁄2 equals 1⁄2.

00:54 So now we have our binomial in the form of x squared plus bx.

00:59 We can now find our b over 2 squared and add it to both sides.

01:02 Normally we would just add it to the binomial, but since this is an equation, what you do to one side, you have to do to both.

01:09 So we're looking for b over two squared.

01:12 Our b, in this case, is going to be our coefficient on our x, which is just one -half.

01:18 So we will have one -half over two squared.

01:23 One -half over two will simplify down to 1 over 4 squared, which you can simplify to 1 over 16.

01:32 So now that we have our b over 2 squared, we can plug that into both sides of our equation.

01:38 So on the left, we'll have x squared minus one -half x plus one over 16 equals one -half plus one over 16.

01:54 Now we can simplify...

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