Solve each equation by completing the square. Write complex...

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Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0. Understanding the structure of quadratic equations is important because it provides the basis for various solution methods such as factoring, using the quadratic formula, or completing the square.

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form (x - h)² + k, which makes it easier to solve by taking the square root of both sides. This method is particularly useful for solving equations and can also aid in understanding the graph of the quadratic function, demonstrating its vertex and axis of symmetry.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part and are expressed in the form a ± bi. They become relevant when solving quadratic equations that yield a negative discriminant, as completing the square in such cases results in an expression involving the square root of a negative number.

Transcript

00:01 We're being asked to solve the given equation using our quadratic formula, which i wrote here off to the right.

00:05 Well, first thing we need to do is set our equation equal to zero.

00:09 So to do that, i'm going to add 2x to both sides of our equation.

00:13 So we're going to be left with 6x squared plus 2x plus 1 is equal to 0.

00:19 And now that our equation is set equal to 0, we can establish our a, b, and c values.

00:24 So for this equation, a will equal to 6, b will be equal to 2, and c will be equal to 1.

00:30 So let's substitute these values into our formula.

00:34 So we're going to have x equals negative 2 plus or minus the square root of 2 squared minus 4 times 6 times 1.

00:43 And it's all getting divided by 2 times 6.

00:46 And now we just need to simplify.

00:48 So we're going to have negative 2 plus or minus.

00:52 Now inside our radical we'll have 2 squared, which is 4.

00:56 And then we have to multiply negative 4 times 6, which is negative 24.

01:00 And negative 24 times 1 is negative 24.

01:05 And it's getting divided by, well, 2 times 6 is 12.

01:08 Well, inside our radical, we have 4 minus 24, which is negative 20.

01:12 So we're going to have negative 2 plus or minus the square root of negative 20, all divided by 12...

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