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

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Key Concepts

Quadratic Equations

Quadratic equations are second-degree polynomial equations typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ? 0. They are central to algebra and can yield two solutions, which may be real or complex depending on the discriminant.

Completing the Square

Completing the square is a method used to transform a quadratic equation into the form (x - h)² = k by adding and subtracting the appropriate constant. This technique simplifies solving the equation by allowing one to take square roots directly, and it is also useful for converting quadratics to vertex form.

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part, typically written in the form a ± bi where i is the square root of -1. They are essential when solving quadratic equations whose discriminants are negative, as such equations do not have real solutions.

Transcript

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

00:06 Well, in order to use it, our equation has to be set equal to zero.

00:09 So the first thing we need to do is add 2x to both sides of our equation.

00:13 So we're going to be left with 4x squared plus 2x plus 3 equals 0.

00:20 Now we can use our quadratic formula.

00:22 So first, let's establish our a, b, and c values.

00:25 Well, a will be equal to 4, b will equal to 2, and c will be equal to 3.

00:29 So let's substitute these values into our formula.

00:33 So when we do, we're going to have x equals negative 2 plus or minus the square root of 2 squared minus 4 times 4 times 3, all divided by 2 times 4.

00:46 And now we just need to simplify.

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

00:52 Well, inside our radical, we have 2 squared, which is 4.

00:55 Then we have negative 4 times 4, which is negative 16.

00:59 And negative 16 times 3 is negative 48.

01:03 And it's all getting divided by 2 times 4, which is 8.

01:06 Well, 4 minus 48 is negative 44.

01:09 We have x equals negative 2 plus or minus the square root of negative 44, all divided by 8...