a Note that the x-intercepts have y-coordinates equal to 0.
B
b Use a calculator to approximate sqrt(6).
A
aSolutions: x=-1+sqrt(6) and x=-1-sqrt(6)
Number of x-intercepts: Two
B
b At x≈ 1.45 and x≈ -3.45
Practice makes perfect
a Before we solve the given equation, we can determine the number of the x-intercepts of the graph of y=x^2+2x-5. These are the points where the y-value of the function is 0. Therefore, the x-intercepts of the graph are the solutions to the given equation.
x^2+2x-5= 0
To determine the number of solutions of a quadratic equation, we will use the discriminant of the given quadratic equation. In the Quadratic Formula, b^2-4ac is the discriminant.
ax^2+bx+c=0
⇕
x=- b±sqrt(b^2-4ac)/2a
Let's identify the values of a, b, and c in our case.
x^2+2x-5=0 ⇕ 1x^2+ 2x+( -5)=0
Now, we can evaluate the discriminant.
Since the discriminant is 24, the quadratic equation has two real solutions. Therefore, the given function has two x-intercepts.
Let's now solve the equation using the Quadratic Formula. Recall that we have already calculated the discriminant, b^2-4ac= 24.
If the discriminant is greater than zero, the equation will have two real solutions. If it is equal to zero, the equation will have one real solution. Finally, if the discriminant is less than zero, the equation will have no real solutions.
b We found that the solutions for the given equation are x=-1±sqrt(6). These are the x-coordinates for which the given function equals 0, so these are the points where the graph crosses the x-axis. Let's use a calculator to approximate sqrt(6).
x=-1±sqrt(6)
x_1=-1+sqrt(6)
x_2=-1-sqrt(6)
x_1=-1+2.449...
x_2=-1-2.449...
x_1=1.449...
x_2=-3.449...
x_1≈1.45
x_2≈ -3.45
Therefore, the graph y=x^2+2x-5 crosses the x-axis at x≈ 1.45 and x≈ -3.45.
