Question

Solve the equation. $$-16 t^{2}+48=0$$

Short answer

The solutions to the equation are \(t = \sqrt{3}\) and \(t = -\sqrt{3}\).

Step-by-step solution

Express the Equation in Standard Form

Start by rewriting the equation in standard form, which is \(at^{2}+c=0\). So, the given equation \(-16t^{2}+48=0\) can be rewritten as \(-16t^{2}= -48\).

Isolate \(t^{2}\)

Divide both sides of the equation by -16 to isolate \(t^{2}\). The equation becomes \(t^{2} = -48 / -16\) which simplifies to \(t^{2} = 3\).

Solve for \(t\)

To solve for \(t\), take the square root of both sides of the equation. Remember to consider both the positive and negative square roots. Therefore, the solutions are \(t = \sqrt{3}\) and \(t = -\sqrt{3}\).

Key concepts

Standard Form of Quadratic Equation

Understanding the standard form of a quadratic equation is the first step in solving such equations. A quadratic equation is typically written as (ax^2 + bx + c = 0), where (a), (b), and (c) are constants, and (a) is not zero. The reason (a) cannot be zero is that it would make the equation linear, not quadratic. This standard form is crucial as it provides a clear structure that allows you to use a variety of methods, like factoring, completing the square, or using the quadratic formula, to find the equation's roots.

When you encounter a quadratic equation that is not in standard form, your first task is to rearrange it. That's exactly what was done in the initial step for the provided exercise, transforming (-16t^2 + 48 = 0) to the standard form by isolating the quadratic term on one side and setting the equation equals to zero, resulting in (-16t^2 = -48).

Isolate the Variable

To solve for a variable means to find the value of the variable that makes the equation true. Isolating the variable is a core technique in algebra. It involves manipulating the equation to get the variable by itself on one side of the equal sign. In the context of our quadratic equation, once we have the standard form, the next step is to isolate the variable term - which, in this case, is (t^2).

For the problem (-16t^2 = -48), we start by isolating (t^2) through division. Dividing both sides by (-16), we obtain (t^2 = 3). This step simplifies the quadratic equation and sets the stage for finding the actual values of (t) using methods suitable for square root extraction.

Square Roots

The square root operation is essential when dealing with quadratic equations. When you have isolated (t^2), as in our exercise (t^2 = 3), finding the square root of both sides is the next step.

Recall that every positive number has two square roots: one positive and one negative. This is because a positive number times itself or its negative counterpart will yield the same positive square value. For instance, both (2) and (-2) squared equal (4). Therefore, when taking the square root of (3) in the equation (t^2 = 3), you must consider both (t = sqrt(3)) and (t = -sqrt(3)) as potential solutions. Keeping this principle in mind is critical to solving quadratic equations accurately, ensuring that you don't miss out on any solutions.