For the inequality: 4𝑥^2 − 28𝑥 + 49 ≤ 0
b) Explain how you could have used the discriminant to find the same answer.
To use the discriminant to find the same answer, we first need to determine the discriminant of the quadratic equation. The discriminant is given by the formula:
Discriminant (D) = b^2 - 4ac
In the given inequality, the quadratic equation is in the form: 4𝑥^2 − 28𝑥 + 49.
Comparing this equation with the standard form of a quadratic equation: 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0, we can identify that:
a = 4
b = -28
c = 49
Now, we can substitute the values into the discriminant formula:
D = (-28)^2 - 4(4)(49)
Simplifying this calculation:
D = 784 - 784
D = 0
If the discriminant is equal to 0, it means that the quadratic equation has a single real root or it is a perfect square trinomial. In the case of the given inequality, it implies that the quadratic expression 4𝑥^2 − 28𝑥 + 49 = 0 has only one solution.
Since the inequality states "≤ 0", we include the endpoint of the solution on the number line. Therefore, we can conclude that the quadratic expression is non-positive for all values of 𝑥, or in other words, the solution to the inequality is the set of all real numbers or (-∞, ∞).