a Recall that an x-intercept is the x-coordinate of the point at which the graph crosses the x-axis.
B
b Recall that an x-intercept is the x-coordinate of the point at which the graph crosses the x-axis.
A
a x ≈ 0.7 or x ≈ 4.3
B
b x ≈ 1.6 or x ≈ - 3.6
Practice makes perfect
a We want to visually estimate the x-intercepts of the given parabola. An x-intercept is the x-coordinate of the point at which the graph intercepts the x-axis. Let's consider the given graph.
The x-intercepts occur at approximately 0.7 and 4.3. Let's now calculate them algebraically to confirm our estimates.
We will use the Quadratic Formula to solve the given quadratic equation.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
Think of the point where the graph of a function crosses the x-axis. The y-value of that ( x, y) coordinate pair is 0, and the x-value is the x-intercept. Thus, to find the x-intercepts of the function we should start by setting the function equal to 0 and identifying the values of a, b, and c.
0=x^2-5x+3 ⇕ 1x^2+( - 5)x+ 3=0
We see that a= 1, b= - 5, and c= 3. Let's substitute these values into the Quadratic Formula.
The solutions for this equation are x= 5± sqrt(13)2. Let's separate them into the positive and negative cases and approximate their values.
x=5± sqrt(13)/2
x_1=5+ sqrt(13)/2
x_2=5- sqrt(13)/2
x_1 ≈ 5+ 3.61/2
x_2 ≈ 5- 3.61/2
x_1 ≈ 8.61/2
x_2 ≈ 1.39/2
x_1 ≈ 4.305
x_2 ≈ 0.695
x_1 ≈ 4.3
x_2 ≈ 0.7
Using the Quadratic Formula, we found that the solutions of the given equation are x_1 ≈ 4.3 and x_2 ≈ 0.7. Notice that we obtained the same values as in our estimations. Thus, our estimations were correct.
b We want to visually estimate the x-intercepts of the given parabola. An x-intercept is the x-coordinate of the point at which the graph intercepts the x-axis. Let's consider the given graph.
The x-intercepts occur at approximately - 3.6 and 1.6. Let's now calculate them algebraically to confirm our estimates.
We will use the Quadratic Formula to solve the given quadratic equation.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
Think of the point where the graph of a function crosses the x-axis. The y-value of that ( x, y) coordinate pair is 0, and the x-value is the x-intercept. Thus, to find the x-intercepts of the function we should start by setting the function equal to 0 and identifying the values of a, b, and c.
0=x^2+2x-6 ⇕ 1x^2+ 2x+( - 6)=0
We see that a= 1, b= 2, and c= - 6. Let's substitute these values into the Quadratic Formula.
The solutions for this equation are x=- 1± sqrt(7). Let's separate them into the positive and negative cases and approximate their values.
x=- 1± sqrt(7)
x_1=- 1+ sqrt(7)
x_2=- 1- sqrt(7)
x_1 ≈ - 1 + 2.645
x_2 ≈ - 1 - 2.645
x_1 ≈ 1.645
x_2 ≈ - 3.645
x_1 ≈ 1.6
x_2 ≈ - 3.6
Using the Quadratic Formula, we found that the solutions of the given equation are x_1 ≈ 1.6 and x_2 ≈ - 3.6. Notice that we obtained the same values as in our estimations. Thus, our estimations were correct.
