Question

\(3 t^2+6 t=18\)

Short answer

The solutions to the equation \(3 t^2+6 t=18\) are \(t = 2\) and \(t = -3\).

Step-by-step solution

Arrange in Standard Form

To transform the equation into the standard form, subtract 18 from both sides of the equation. Thus the equation becomes: \(3t^2 + 6t - 18 = 0\)

Factoring

At this stage, the quadratic equation can be factored. Remember that in a quadratic equation of the form \(at^2 + bt + c = 0\), the factored form will look like \((t-p)(t-q)=0\). Factoring the quadratic equation leads to \((t - 2)(3t + 9) = 0\)

Solve for \(t\)

Having factored the equation, now set each group of parentheses equal to zero, and solve for \(t\). This will give you the value(s) of \(t\) that satisfy the equation. Solving gives \(t = 2\) and \(t = -3\)

Key concepts

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is a vital foundation for solving such equations. It is presented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). The standard form makes it clear that a quadratic equation is defined by three terms: the quadratic term \(ax^2\), the linear term \(bx\), and the constant term \(c\).

To solve a quadratic equation, it often needs to be rearranged into this standard form. This helps in applying various methods for finding its solutions, such as factoring, completing the square, or using the quadratic formula. In the example \(3t^2+6t=18\), rewriting it as \(3t^2 + 6t - 18 = 0\) puts it in the standard form, setting the stage for factoring and solution extraction.

Factoring Quadratic Equations

Factoring is a method used to solve quadratic equations when they can be expressed as the product of two binomials. In the context of the equation \(3t^2 + 6t - 18 = 0\), factoring involves finding two numbers that both add to the middle term coefficient (in this case, 6) and multiply to give the product of the quadratic coefficient and the constant term (in this case, \(3 \times (-18)\)).

Once the equation is factored, as shown \(3t^2 + 6t - 18 = (t - 2)(3t + 9) = 0\), this factored form can be used to find the solutions of the equation. The principle behind this method is the zero product property, which states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property allows us to set each factor equal to zero and solve for the variable.

Quadratic Equation Solutions

The solutions to a quadratic equation are the values that satisfy the equation when plugged in for the unknown variable. In the provided example, factoring the quadratic equation \(3t^2 + 6t - 18 = 0\) produces \(t - 2)(3t + 9) = 0\). Applying the zero product property gives us two potential solutions: \(t = 2\) and \(t = -3\).

Quadratic equations can have either two real solutions, one real solution, or two complex solutions. The number of real solutions is determined by the discriminant, \(b^2 - 4ac\). If the discriminant is greater than zero, there are two distinct real solutions; if it is equal to zero, there is one real solution (a repeated solution); and if it is less than zero, the solutions are complex. The solutions we found here are both real and distinct, satisfying the original equation.