Question

Solve by quadratic formula. Give your answers in decimal form to three significant digits. Check some by calculator. $$5 x^{2}-25 x+4=0$$

Short answer

\(x \approx 4.866\) and \(x \approx 0.1639\). These are the solutions to the given quadratic equation in decimal form, rounded to three significant digits.

Step-by-step solution

Identify Coefficients

For the quadratic equation of the form \(ax^{2} + bx + c = 0\), identify the coefficients. In this case, \(a = 5\), \(b = -25\), and \(c = 4\).

Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) to find the roots of the equation. Substitute \(a = 5\), \(b = -25\), and \(c = 4\) into the formula.

Calculate the Discriminant

Calculate the discriminant \(\Delta = b^2 - 4ac\) using the values obtained in Step 1. This will be \(\Delta = (-25)^2 - 4(5)(4)\).

Evaluate the Quadratic Formula

Evaluate the plus and minus parts of the quadratic formula separately to find the two possible values for \(x\).

Simplify and Find Decimal Values

Simplify the results from Step 4 to get them in decimal form to three significant digits.

Check with Calculator

Use a calculator to confirm that the decimal values obtained are correct to three significant digits.

Key concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form ax² + bx + c = 0, where a, b, and c are coefficients, and x represents the variable. To solve such equations and find the values of x, we often use the quadratic formula, which is a universal method providing a straightforward solution process.

The formula itself is given by x = \( \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \), where the ± symbol indicates that there will generally be two solutions, known as the roots of the equation. These roots can be real or complex numbers depending upon the discriminant (which we will discuss in the next section). Once you insert the coefficients into this formula and simplify, you can find the solutions that make the original quadratic equation true.

It's important for students to be comfortable with each step in applying the quadratic formula. This includes understanding how to identify coefficients, perform operations under the radical (the square root symbol), and simplify their results, which is an essential skill in algebra.

Discriminant Calculation

The discriminant in a quadratic equation is a critical number that tells us about the nature of the roots without actually solving the equation. It is represented by the Greek letter Delta (\( \Delta \)) and is calculated as \( \Delta = b^2 - 4ac \). Here, b and c again are the coefficients from the quadratic equation ax² + bx + c = 0.

The value of the discriminant provides three possible scenarios: If \( \Delta > 0 \), there are two distinct real roots. If \( \Delta = 0 \), there is exactly one real root (also called a repeated or double root). If \( \Delta < 0 \), the roots are complex and have imaginary numbers as part of the solution.

Calculating the discriminant is a step that should not be overlooked because it gives insight into the results we can expect before computing the actual roots. This also helps in understanding why some quadratic equations produce real numbers while others give us complex numbers as solutions.

Significant Digits

Significant digits, or significant figures, represent the precision of a numerical answer. They are crucial in mathematics and sciences because they convey how accurate a number is. When we solve for the roots of a quadratic equation, especially when using a calculator, we often get a long string of digits. However, in many cases, only a certain number of those digits carry meaning.

The rules for determining significant digits can be summarized as follows: Non-zero digits are always significant. Any zeros between non-zero digits are significant. Leading zeros are not significant. Trailing zeros are significant only if the number contains a decimal point.

When an exercise specifies the use of a certain number of significant digits in the solution, we must round our answer to that many digits while keeping the value as precise as possible. This process eliminates extraneous information and reduces the likelihood of carrying forward small rounding errors in subsequent calculations. As such, mastering significant digits can greatly enhance the clarity and reliability of numerical answers.