(d) The Equation X 2 − X − 1 = 0 X^2 - X - 1 = 0 X 2 − X − 1 = 0 Can Be Solved By Drawing A Straight Line On The Graph Of Y = X 2 + X − 3 Y = X^2 + X - 3 Y = X 2 + X − 3 .(i) Find The Equation Of This Straight Line. Answer: __________(ii) Draw This Straight Line And Hence Solve $x^2 - X -

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. While traditional methods such as factoring and the quadratic formula are widely used, graphical methods can also be employed to solve quadratic equations. In this article, we will explore how to solve the quadratic equation $x^2 - x - 1 = 0$ by drawing a straight line on the graph of $y = x^2 + x - 3$.

Understanding the Problem

The given quadratic equation is $x^2 - x - 1 = 0$. To solve this equation using graphical methods, we need to find the equation of a straight line that intersects the graph of $y = x^2 + x - 3$ at the points where the quadratic equation is satisfied. This means that the straight line will have a slope and a y-intercept that will allow it to intersect the graph of the quadratic equation at the desired points.

Finding the Equation of the Straight Line

To find the equation of the straight line, we need to determine its slope and y-intercept. The slope of the straight line can be found by considering the difference in the y-coordinates of the points where the quadratic equation is satisfied. Let's assume that the straight line intersects the graph of the quadratic equation at the points $(x_1, y_1)$ and $(x_2, y_2)$. The slope of the straight line can be calculated as:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

However, since we are not given the exact points of intersection, we need to use a different approach to find the equation of the straight line. We can start by rewriting the quadratic equation as:

$$x^2 - x - 1 = 0$$

$$x^2 - x = 1$$

$$x(x - 1) = 1$$

This equation can be rewritten as:

$$x^2 - x - 1 = 0$$

$$x^2 - x = 1$$

$$x^2 - x + \frac{1}{4} = 1 + \frac{1}{4}$$

$$(x - \frac{1}{2})^2 = \frac{5}{4}$$

$$x - \frac{1}{2} = \pm \sqrt{\frac{5}{4}}$$

$$x = \frac{1}{2} \pm \frac{\sqrt{5}}{2}$$

Now, we can substitute these values of $x$ into the equation of the quadratic function to find the corresponding values of $y$:

$$y = x^2 + x - 3$$

$$y = (\frac{1}{2} \pm \frac{\sqrt{5}}{2})^2 + (\frac{1}{2} \pm \frac{\sqrt{5}}{2}) - 3$$

Simplifying the expression, we get:

$$y = \frac{1}{4} \pm \frac{\sqrt{5}}{2} + \frac{1}{2} \pm \frac{\sqrt{5}}{2} - 3$$

$$y = -\frac{5}{4} \pm \sqrt{5}$$

Now, we can see that the straight line intersects the graph of the quadratic equation at the points $(\frac{1}{2} \pm \frac{\sqrt{5}}{2}, -\frac{5}{4} \pm \sqrt{5})$. The slope of the straight line can be calculated as:

$$m = \frac{-\frac{5}{4} + \sqrt{5} - (-\frac{5}{4} - \sqrt{5})}{\frac{1}{2} + \frac{\sqrt{5}}{2} - (\frac{1}{2} - \frac{\sqrt{5}}{2})}$$

$$m = \frac{2\sqrt{5}}{\sqrt{5}}$$

$$m = 2$$

The y-intercept of the straight line can be found by substituting $x = 0$ into the equation of the straight line:

$$y = 2(0) + b$$

$$y = b$$

Since the straight line intersects the graph of the quadratic equation at the points $(\frac{1}{2} \pm \frac{\sqrt{5}}{2}, -\frac{5}{4} \pm \sqrt{5})$, we can substitute these values of $x$ and $y$ into the equation of the straight line to find the value of $b$:

$$-\frac{5}{4} \pm \sqrt{5} = 2(\frac{1}{2} \pm \frac{\sqrt{5}}{2}) + b$$

Simplifying the expression, we get:

$$-\frac{5}{4} \pm \sqrt{5} = 1 \pm \sqrt{5} + b$$

$$b = -\frac{5}{4} \pm \sqrt{5} - 1 \pm \sqrt{5}$$

$$b = -\frac{9}{4}$$

Therefore, the equation of the straight line is:

$$y = 2x - \frac{9}{4}$$

Drawing the Straight Line and Solving the Quadratic Equation

To draw the straight line, we can use the equation $y = 2x - \frac{9}{4}$. We can plot the points $(\frac{1}{2} \pm \frac{\sqrt{5}}{2}, -\frac{5}{4} \pm \sqrt{5})$ on the graph and draw a straight line through these points. The straight line will intersect the graph of the quadratic equation at the points where the quadratic equation is satisfied.

To solve the quadratic equation, we can use the equation of the straight line to find the values of $x$ that satisfy the equation. We can substitute the equation of the straight line into the equation of the quadratic function to get:

$$x^2 + x - 3 = 2x - \frac{9}{4}$$

Simplifying the expression, we get:

$$x^2 - x + \frac{9}{4} = 0$$

This equation can be rewritten as:

$$(x - \frac{1}{2})^2 = \frac{9}{16}$$

$$x - \frac{1}{2} = \pm \frac{3}{4}$$

$$x = \frac{1}{2} \pm \frac{3}{4}$$

Therefore, the solutions to the quadratic equation are:

$$x = \frac{1}{2} + \frac{3}{4} = \frac{5}{4}$$

$$x = \frac{1}{2} - \frac{3}{4} = -\frac{1}{4}$$

Conclusion

$$$$$$$$$$

Q: What is the main idea behind solving quadratic equations through graphical methods?

A: The main idea behind solving quadratic equations through graphical methods is to use the graph of the quadratic function to find the solutions to the equation. This is done by drawing a straight line on the graph that intersects the quadratic function at the points where the equation is satisfied.

Q: How do I find the equation of the straight line that intersects the quadratic function?

A: To find the equation of the straight line, you need to determine its slope and y-intercept. The slope of the straight line can be found by considering the difference in the y-coordinates of the points where the quadratic equation is satisfied. The y-intercept of the straight line can be found by substituting x = 0 into the equation of the straight line.

Q: What is the significance of the points of intersection between the straight line and the quadratic function?

A: The points of intersection between the straight line and the quadratic function are the solutions to the quadratic equation. By finding the equation of the straight line, you can determine the values of x that satisfy the equation.

Q: How do I draw the straight line on the graph of the quadratic function?

A: To draw the straight line, you can use the equation of the straight line to plot the points of intersection between the straight line and the quadratic function. You can then draw a straight line through these points to visualize the solutions to the equation.

Q: What are the advantages of solving quadratic equations through graphical methods?

A: The advantages of solving quadratic equations through graphical methods include:

• Visualizing the solutions to the equation
• Understanding the relationship between the quadratic function and the straight line
• Developing problem-solving skills through graphical analysis
• Enhancing mathematical literacy and critical thinking

Q: What are the limitations of solving quadratic equations through graphical methods?

A: The limitations of solving quadratic equations through graphical methods include:

• Difficulty in visualizing the graph for complex equations
• Limited accuracy in determining the points of intersection
• Time-consuming process of drawing the graph and finding the equation of the straight line

Q: Can I use graphical methods to solve quadratic equations with complex coefficients?

A: While graphical methods can be used to solve quadratic equations with complex coefficients, it may be more challenging to visualize the graph and determine the points of intersection. In such cases, it may be more practical to use traditional methods such as factoring or the quadratic formula.

Q: Can I use graphical methods to solve quadratic equations with multiple solutions?

A: Yes, graphical methods can be used to solve quadratic equations with multiple solutions. By drawing the graph of the quadratic function and the straight line, you can visualize the multiple solutions to the equation.

Q: How can I apply graphical methods to solve quadratic equations in real-world problems?

A: Graphical methods can be applied to solve quadratic equations in real-world problems such as:

• Modeling population growth and decline
• Analyzing the motion of objects under the influence of gravity
• Determining the maximum or minimum value of a function
• Solving optimization problems in business and economics

By applying graphical methods to solve quadratic equations, you can develop a deeper understanding of the mathematical concepts and their applications in real-world problems.