Solve #x^2-x-2=0# graphically, by first sketching #y=x^2#?

Question

Elijah Allen · Accepted Answer

#x=-1" or " x=2# #color(blue)"sketching " y=x^2# #"given a quadratic in standard form " ax^2+bx+c# #• " if " a>0" then graph is a minimum " uuu# #• " if " a<0" then graph is a maximum " nnn# #"the coordinates of the vertex are " (-b/(2a),f(-b/(2a)))# #"the y-intercept is the value of the constant c"# #"the x-intercepts are found by equating to zero"# #"for " y=x^2# #a>0rArruuu# #"the coordinates of the vertex are " (0,0)# #"the value of " c=0# #"some points on the graph "# #x=+-1toy=1rArr(1,1),(-1,1)# #x=+-2toy=4rArr(2,4),(-2,4)" etc"# graph{x^2 [-10, 10, -5, 5]} #color(blue)"sketching " y=x^2-x-2# #• " for coefficient of x " >0# #"the vertex moves " -b/(2a)larr" to the left"# #• " for coefficient of x " <0# #"the vertex moves " -b/(2a)rarr" to the right"# #y=x^2-x-2 " is the same shape as " y=x^2# #"coefficient of x term is " -1# #rArrx_(color(red)"vertex")=--1/(2)=1/2rarr# #rArry_(color(red)"vertex")=(1/2)^2-1/2-2=-9/4# #rArrcolor(magenta)"vertex "=(1/2,-9/4)# #"the y-intercept is at " c=-2rArr(0,-2)# #"we could find the solution to " x^2-x-2" algebraically"# #"by solving " x^2-x-2=0# #"However, from the graph the solutions are the values "# #"of x where the graph crosses the x-axis"# #"that is " x=-1" or " x=2# graph{x^2-x-2 [-10, 10, -5, 5]}

Madelyn Anderson · Answer

undefined To solve the equation $x^2 - x - 2 = 0$ graphically, we first sketch the graph of $y = x^2$, which is a parabola.

Now, let's plot the graph of $y = x^2$. This is a simple quadratic function, and its graph is a parabola that opens upwards.

To sketch the graph, we'll plot a few points and then connect them to get the shape of the parabola. We choose a range of x-values and calculate the corresponding y-values by squaring each x-value.

Let's take some arbitrary x-values, say -2, -1, 0, 1, and 2:

When $x = -2$, $y = (-2)^2 = 4$
When $x = -1$, $y = (-1)^2 = 1$
When $x = 0$, $y = (0)^2 = 0$
When $x = 1$, $y = (1)^2 = 1$
When $x = 2$, $y = (2)^2 = 4$

Plotting these points, we get (-2, 4), (-1, 1), (0, 0), (1, 1), and (2, 4). Connecting these points gives us a parabola that opens upwards.

Now, to solve $x^2 - x - 2 = 0$ graphically, we need to find the points where the graph of $y = x^2$ intersects the x-axis. These points represent the solutions to the equation $x^2 - x - 2 = 0$, where the curve intersects the x-axis.

We'll look for the x-values where $y = x^2$ intersects the x-axis, indicating where $x^2 - x - 2 = 0$.

By inspection, we see that the curve intersects the x-axis at approximately $x = -1$ and $x = 2$.

Therefore, the solutions to the equation $x^2 - x - 2 = 0$ are $x = -1$ and $x = 2$.