For the inequality: 4𝑥^2 − 28𝑥 + 49 ≤ 0
a) Solve by either the graphing or the number-line method.
b) Explain how you could have used the discriminant to find the same answer.
a) To solve the inequality by graphing, we need to first convert it to an equation: 4𝑥^2 − 28𝑥 + 49 = 0.
To find the roots of the equation, we can factor it as (2𝑥 - 7)(2𝑥 - 7) = 0, which gives us the double root 𝑥 = 7/2.
Since the inequality is less than or equal to, we include the double root in our solution.
Now we can plot the graph of the function y = 4𝑥^2 − 28𝑥 + 49 and observe where it is less than or equal to zero.
The graph is a upward-opening parabola, and since the coefficient of 𝑥^2 is positive, the parabola opens upwards.
Knowing that the x-coordinate of the vertex is given by the formula 𝑥 = -𝑏/2𝑎, we can compute that 𝑥 = -(-28)/2(4) = 7/2, which matches the root we found earlier.
Therefore, the graph touches the x-axis at 𝑥 = 7/2 and continues to open upwards.
From the graph, we can see that the function is less than or equal to zero in the interval (-∞, 7/2].
This means that the solution to the inequality is 𝑥 ≤ 7/2.
b) The discriminant can be used to find the same answer by determining the nature of the roots of the equation.
The discriminant is given by 𝑏^2 - 4𝑎𝑐 for a quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0.
In our equation 4𝑥^2 − 28𝑥 + 49 = 0, the values of 𝑎, 𝑏, and 𝑐 are 4, -28, and 49 respectively.
Substituting these values into the discriminant formula, we have:
𝑏^2 - 4𝑎𝑐 = (-28)^2 - 4(4)(49) = 784 - 784 = 0.
Since the discriminant is equal to zero, it means that the equation has a double root.
This confirms what we found earlier when we factored the equation, where we obtained the double root 𝑥 = 7/2.
Therefore, the solution to the inequality is 𝑥 ≤ 7/2.