Problem 53 - Algebra and Trigonometry

Question

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ x^{2}-0.011 x-0.064=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 0.259 \) and \( x = -0.248 \).

Step 1: Identify Coefficients

For the quadratic equation in the form \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = 1 \), \( b = -0.011 \), and \( c = -0.064 \).

Step 2: Write the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the values of \( x \).

Step 3: Calculate the Discriminant

The discriminant \( \Delta \) is \( b^2 - 4ac \). Substitute the values: \( \Delta = (-0.011)^2 - 4 \times 1 \times (-0.064) \). Calculate \( \Delta = 0.000121 + 0.256 = 0.256121 \).

Step 4: Solve for the Roots

Using the quadratic formula, substitute the values of \( a \), \( b \), and \( \Delta \):\[ x = \frac{-(-0.011) \pm \sqrt{0.256121}}{2 \times 1} \]\[ x = \frac{0.011 \pm 0.506085}{2} \]Calculate each root separately:For \( x_1 = \frac{0.011 + 0.506085}{2} = 0.2585 \)For \( x_2 = \frac{0.011 - 0.506085}{2} = -0.2475 \)

Step 5: Round the Solutions

Round the solutions to three decimal places:\( x_1 = 0.259 \), \( x_2 = -0.248 \).

Key Concepts

Quadratic Equation

A quadratic equation is an essential concept in algebra that appears in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The goal is to find values of \( x \) that satisfy the equation.Typically, a quadratic equation represents a parabola on a graph, which can either open upwards or downwards. This shape depends on whether the value of \( a \) is positive or negative. Understanding quadratic equations is important for a range of mathematical applications and real-world problems.

• The "\( ax^2 \)" term gives it a parabolic shape.
• The "\( b \)" coefficient affects the position and orientation.
• The "\( c \)" constant helps determine the y-intercept.

By finding the roots of this equation, we are essentially determining where the parabola intersects the x-axis, if at all.

Real Solutions

Real solutions are those solutions to the quadratic equation that are real numbers, meaning they can be expressed as a count or measure on the number line. Not all quadratic equations have real solutions. In some cases, the solutions might be complex numbers, which involve imaginary numbers. However, if an equation has real solutions, it means that its graph intersects the x-axis at real number points. In this context, solutions can be:

• Two distinct real solutions (parabola crosses the x-axis twice).
• One real solution (parabola just touches the x-axis).
• No real solution (parabola does not intersect the x-axis).

The presence of real solutions significantly impacts how the quadratic equation is utilized in various scenarios, from science to engineering.

Discriminant

The discriminant is a particular value calculated from a quadratic equation and is vital for determining the nature of its solutions. The formula for the discriminant, typically denoted as \( \Delta \), is \( b^2 - 4ac \). This nifty bit of algebra tells us crucial information about the roots:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), there is one real root, technically two identical roots.
• If \( \Delta < 0 \), the roots are not real; they're complex with imaginary parts.

For the provided exercise, the discriminant calculation \( \Delta = 0.256121 \) is greater than zero, indicating the existence of two real roots. Understanding the discriminant's role is crucial in predicting the nature of solutions without solving the equation completely.

Roots of Equations

The roots of equations refer to the solutions that satisfy the equation, generally where the graph of the equation intersects the x-axis.By finding these roots, we uncover the values for \( x \) that make the whole equation zero at that point.To find roots using the quadratic formula, you'll apply:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is essential for solving quadratic equations because it offers a way to find roots systematically and efficiently. In this exercise, substituting the values, we've calculated:

• Root \( x_1 = 0.259 \)
• Root \( x_2 = -0.248 \)

Each solution represents a point at which the quadratic graph touches or crosses the x-axis, providing insights into the behavior of the function in question.