Question

If the roots of the quadratic equation \(2 \mathrm{x}^{2}-5 \mathrm{x}+2=0\) are the intercepts made by a line on the coordinate axes, then the equation of the line can be (1) \(4 x+y=2\) (2) \(2 \mathrm{x}+5 \mathrm{y}+2\) (3) \(x+4 y=2\) (4) Both (1) and (3)

Short answer

Answer: \(x + 4y = 2\)

Step-by-step solution

Find the roots of the quadratic equation

The given quadratic equation is \(2x^2-5x+2=0\). Let's solve for the roots (x-intercepts) using the quadratic formula: \(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\) where a = 2, b = -5, and c = 2.

Calculate the discriminant

Before plugging these values into the quadratic formula, we need to find the discriminant, i.e., the value under the square root: \(D = b^2 - 4ac = (-5)^2 - 4(2)(2) = 25 - 16 = 9\)

Find the roots using the quadratic formula

Now we can find the two roots: \(x_1 = \frac{-(-5)+\sqrt{9}}{2(2)} = \frac{5+3}{4} = \frac{8}{4} = 2\) \(x_2 = \frac{-(-5)-\sqrt{9}}{2(2)} = \frac{5-3}{4} = \frac{2}{4} = \frac{1}{2}\)

Use the roots as the intercepts for the line equation

Now that we have the roots (x-intercepts), we can find the corresponding y-intercepts as these roots represent the intercepts made by a line on the coordinate axes. For \(x_1=2\), the point on the coordinate axis where the line intersects the x-axis is \((2,0)\). For \(x_2=\frac{1}{2}\), the point on the coordinate axis where the line intersects the y-axis is \(\left(0,\frac{1}{2}\right)\).

Use the two-point form of a line equation

To derive the line equation using the two points we found in step 4, we use the two-point form: \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) Plugging in the coordinate points \((2,0)\) and \(\left(0,\frac{1}{2}\right)\): \(y - 0= \frac{\frac{1}{2} - 0}{0 - 2}(x - 2)\) Simplify the equation: \(y = \frac{-\frac{1}{2}}{-2} (x - 2)\) \(y = \frac{1}{4}(x - 2)\)

Convert to standard form and compare with the given options

To identify which of the given options match our derived equation, let's convert our equation to its standard form: \(4y = x - 2\) \(4y - x = -2\) or \(x - 4y = 2\) Matching our derived line equation with the given options, we can conclude that the correct option is: (3) \(x+4 y=2\) Hence, the equation of the line with intercepts as the roots of the quadratic equation is \(x + 4y = 2\).

Key concepts

Roots of a Quadratic Equation

Quadratic equations are of the form \(ax^2 + bx + c = 0\). The solutions to this equation, known as the roots, are the values of \(x\) that make the equation true. Understanding the roots is crucial, as they often represent significant points on a graph, like where the graph crosses the x-axis. Roots can be real or complex numbers. Real roots occur when the discriminant \(b^2 - 4ac\) is greater than or equal to zero, while complex roots occur when this discriminant is negative. This particular equation \(2x^2 - 5x + 2 = 0\) has a positive discriminant of 9, indicating two distinct real roots. These roots are determined using the quadratic formula, which provides a clear route to calculating them efficiently.

Two-Point Form of a Line

The two-point form is a method used to find the equation of a line when two points on the line are known. This form can be expressed as:

• \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\)

In this formula, \((x_1, y_1)\) and \((x_2, y_2)\) represent two points on the line. By substituting these points into the equation, one can derive the equation of the line that connects them. For instance, with intercepts \( (2,0) \) and \( \left(0,\frac{1}{2}\right) \), the resulting line equation describes how the line intersects the x and y axes. Utilizing the two-point form is a straightforward approach to determine line equations and can simplify the process of finding an intercept line equation.

Using Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. Its expression, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), allows us to calculate the roots by plugging in the values of \(a\), \(b\), and \(c\). This formula accommodates all possible scenarios regarding the number and nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a repeated root).
• If \(b^2 - 4ac < 0\), the roots are complex and appear as conjugates.

For the equation \(2x^2 - 5x + 2 = 0\), using the quadratic formula reveals roots at \(x = 2\) and \(x = \frac{1}{2}\), which are essential for constructing line equations in other contexts.

Line Equations

Line equations are mathematical representations that describe straight lines in a coordinate plane. The role of line equations underlies the connection between algebra and geometry, helping to translate between the visual information of graphs and numeric computations. Natively, line equations appear in several standard forms, including slope-intercept \(y = mx + b\), point-slope \(y - y_1 = m(x - x_1)\), and standard form \(Ax + By = C\). These forms are interconvertible.In solving real-world problems, often the information provided requires using specific forms using known variables. For example, using intercept points from a quadratic equation as line intercepts allows us to derive the line equation using the two-point form and then convert it into a standard format for direct comparison against given choices. This application further accentuates the utility of understanding line equations.