factor : 4𝑥^2 − 28𝑥 + 49 ≤ 0
To factor the expression 4𝑥^2 − 28𝑥 + 49 ≤ 0, we can use the quadratic formula.
The quadratic formula is given by 𝑥 = (−𝑏 ± √(𝑏^2 − 4𝑎𝑐)) / (2𝑎), where 𝑎 = 4, 𝑏 = −28, and 𝑐 = 49.
Calculating the discriminant, 𝑏^2 − 4𝑎𝑐, we get 784 − 4(4)(49) = 784 − 784 = 0.
Since the discriminant is zero, there is only one root, which means the quadratic expression has no real roots. Therefore, we cannot factor the expression as real factors.
However, we can still determine the solution by the sign of the quadratic expression. When the discriminant is zero, the quadratic expression is either always positive or always negative.
Since the coefficient of 𝑥^2 is positive (4 > 0), the quadratic expression opens upwards, meaning it is always positive.
Therefore, the inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0 has no solution.