Question

Multiply each side of the equation by an appropriate power of ten to obtain integer coefficients. Then solve by factoring.$$0.23 t^{2}-0.54 t+0.16=0$$

Short answer

The solution to the equation is \(t = 4/23\) and \(t = 4\).

Step-by-step solution

Multiply Each Side by Power of Ten

In order to transform decimal coefficients into integers, we need to multiply each side by an appropriate power of ten. Here, the smallest decimal place is two (0.23, 0.54, 0.16), so we multiply the equation by \(10^{2} = 100\). This results in \(23t^{2} - 54t + 16 = 0\).

Factor the Quadratic

Next, we need to factor the equation \(23t^{2} - 54t + 16 = 0\). This can be factored to \((23t - 4)(t - 4) = 0\)

Solve for t

The equation can be solved by setting each factor equal to zero. Therefore, \(23t - 4 = 0\) and \(t - 4 = 0\), gives us \(t = 4/23\) and \(t = 4\).

Key concepts

Factoring Quadratics

Quadratic equations, which are polynomial equations of the second degree, can often seem intimidating, but breaking them down can reveal more straightforward solutions. Factoring is a method to solve these equations and involves expressing the quadratic in the form \(ax^2 + bx + c = 0\) as a product of its factors. To do this, we look for two binomials \( (dx + e)(fx + g) = 0 \) that when multiplied together, give us the original quadratic equation. The values of \(d\), \(e\), \(f\), and \(g\) are found by considering the factors of 'a' and 'c', and how their products, when combined, can yield 'b'.

Why is this useful? Once expressed in factored form, the Zero Product Property states that if a product equals zero, then at least one of the factors must be zero. So, we can set each of the binomial factors equal to zero and solve for the variable. This method effectively breaks down a more complex equation into simpler components that are easier to solve.

Multiplying by Powers of Ten

When encountering quadratic equations with decimal coefficients like \(0.23t^2 - 0.54t + 0.16 = 0\), multiplying each term by powers of ten is a neat trick to simplify the numbers. This technique converts unwieldy decimals into whole numbers, making the equation much more approachable for methods like factoring. The power of ten needed is typically determined by the smallest decimal place in the given coefficients—in our example, this was the hundredths place, so \(10^2\) or 100 was used.

This step transforms the equation to use integers which are easier to factor. Not only does it simplify the equation visually, but it also aids in avoiding common errors when dealing with fractions or decimals. After the multiplication, the quadratic equation's properties remain unchanged, yet dealing with whole numbers tends to be less error-prone, especially for factoring.

Quadratic Formula

When factoring a quadratic equation is too complex or downright impossible, the quadratic formula provides a reliable alternative. The formula takes the form \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) and is applicable to any quadratic equation \(ax^2 + bx + c = 0\). It is rooted in the method of completing the square and offers solutions by accounting for all the coefficients of the equation.

The beauty of the quadratic formula lies in its ability to provide answers regardless of the quadratic equation's factorability. The term under the square root, \(b^2 - 4ac\), known as the discriminant, can also give insight into the nature of the roots—whether they are real or complex, and if they are distinct or repeated. This elegant formula ensures that no quadratic equation has to be left unsolved.