Question

For the inequality: 4𝑥^2 − 28𝑥 + 49 ≤ 0
b) Explain how you could have used the discriminant to find the same answer

Answers

Answered by oobleck
for the quadratic 4𝑥^2 − 28𝑥 + 49 ≤ 0, the discriminant is zero.
so the only solution is the single root x = 7/2
Answered by GPT 3.5
To use the discriminant to find the answer for the inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0, we need to first understand what the discriminant is.

The discriminant is a mathematical term used in quadratic equations to determine the nature of the roots. In a quadratic equation of the form 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0, the discriminant is given by the formula Δ = 𝑏^2 - 4𝑎𝑐.

The value of the discriminant can help us determine whether the quadratic equation has two real, distinct roots (Δ > 0), one real root (Δ = 0), or no real roots (Δ < 0).

In terms of the given inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0, we can use the discriminant to find the same answer by applying the following steps:

1. Set the quadratic equation equal to zero by subtracting 49 from both sides: 4𝑥^2 − 28𝑥 + 49 - 49 ≤ 0, which simplifies to 4𝑥^2 − 28𝑥 ≤ 0.

2. Now that the equation is in the standard quadratic form, we can identify 𝑎, 𝑏, and 𝑐. In this case, 𝑎 = 4, 𝑏 = -28, and 𝑐 = 0.

3. Calculate the discriminant using the formula Δ = 𝑏^2 - 4𝑎𝑐. Plugging in the values, we have Δ = (-28)^2 - 4(4)(0).

4. Simplify the expression: Δ = 784 - 0.

5. Determine the value of the discriminant: Δ = 784.

6. Since the discriminant (Δ) is greater than zero (Δ > 0), this means that the quadratic equation has two real, distinct roots.

However, we are interested in solving the inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0, which means we want to find the values of 𝑥 that make the expression less than or equal to zero.

7. Since the quadratic equation has two real, distinct roots, we can use the concept of concavity to determine the solution to the inequality.

8. Since the coefficient of the leading term (𝑎) is positive (4 > 0), the graph of the quadratic equation will be concave up. This means that the graph will be below or touching the x-axis between the two real roots.

9. Therefore, the solution to the inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0 is all the values of 𝑥 between and including the two real roots of the quadratic equation.

By using the discriminant, we can determine the nature of the roots and use that information to find the solution to the inequality.
Answered by GPT 3.5
Apologies for the incorrect information in my previous response. You are correct.

The discriminant of the quadratic equation 4𝑥^2 − 28𝑥 + 49 ≤ 0 is indeed zero. This means that the equation has only one real root.

The discriminant is calculated using the formula Δ = 𝑏^2 - 4𝑎𝑐. Plugging in the values, we have Δ = (-28)^2 - 4(4)(49).

Simplifying, Δ = 784 - 784 = 0.

Since the discriminant is zero (Δ = 0), this means that the quadratic equation has only one real root.

To find this root, we can use the quadratic formula:

𝑥 = (-𝑏 ± √Δ) / (2𝑎)

In this case, 𝑎 = 4, 𝑏 = -28, and Δ = 0.

𝑥 = (-(-28) ± √0) / (2*4)

Simplifying further:

𝑥 = (28 ± 0) / 8

𝑥 = 28 / 8

𝑥 = 7/2

Therefore, the only solution to the inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0 is 𝑥 = 7/2.

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