Equation that has no real solution: Select the quadratic 9x^2 – 25x + 30 = 0 9x^2 – 30x + 25 = 0 9x^2 + 30x + 25 = 0
The Correct Answer and Explanation is:
The quadratic equation that has no real solution is 9×2−25x+30=09x^2 – 25x + 30 = 0.
To determine which equation has no real solution, we use the discriminant formula from the quadratic equation, which is given by:
Δ=b2−4ac\Delta = b^2 – 4ac
where aa, bb, and cc are the coefficients from the equation ax2+bx+c=0ax^2 + bx + c = 0. If the discriminant is negative, the quadratic equation has no real solution.
For 9×2−25x+30=09x^2 – 25x + 30 = 0:
- a=9a = 9
- b=−25b = -25
- c=30c = 30
The discriminant calculation is:
(−25)2−4(9)(30)=625−1080=−455(-25)^2 – 4(9)(30) = 625 – 1080 = -455
Since the discriminant is negative, the equation has no real solution.
Explanation
Quadratic equations can have real or complex solutions depending on their discriminant. The discriminant is a key factor in the quadratic formula:
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
The value under the square root, b2−4acb^2 – 4ac, determines the nature of the roots:
- If the discriminant is positive, the equation has two distinct real solutions.
- If the discriminant is zero, the equation has one real solution.
- If the discriminant is negative, the equation has no real solutions and instead yields complex numbers.
For the given equations, only 9×2−25x+30=09x^2 – 25x + 30 = 0 has a negative discriminant, making it the correct answer. This means the quadratic expression does not cross the x-axis when plotted, reinforcing the fact that it has no real roots.
