If the roots α and β of the quadratic equation 9x2 - 3x + (m - 4) =

1.	If the roots α and β of the quadratic equation 9x2 - 3x + (m - 4) = 0 are such that α - β = 1, then the value of m is:1. 22. 13. 34. 4
Answer» Correct Answer - Option 1 : 2 Concept: If α and β are the two roots of the quadratic polynomial Ax² + Bx + C, then: Ax² + Bx + C = (x - α)(x - β) = x² - (α + β)x + αβ = 0 α + β = \(\rm -\frac{B}{A}\) and αβ = \(\rm \frac{C}{A}\). Calculation: Comparing the given equation 9x2 - 3x + (m - 4) = 0 with the general equation Ax2 + Bx + C = 0, we can say that: A = 9, B = -3, C = (m - 4) It is given that: α - β = 1 ... (1) Using α + β = \(\rm -\frac{B}{A}\), we get: α + β = \(\rm -\left(\frac{-3}{9}\right)=\frac13\) ... (2) Adding equations (1) and (2), we get: \(\rm 2\alpha = 1+\frac13=\frac43\) ⇒ \(\rm \alpha = \frac23\) Using equation (2), we get: \(\rm \beta = \frac13-\frac23=-\frac13\) Now, the product of the roots is: αβ = \(\rm \frac{C}{A}\). ⇒ \(\rm \left(\frac23\right)\times\left(-\frac13\right)=\frac{m-4}{9}\) ⇒ m - 4 = -2 ⇒ m = 2.

If the roots α and β of the quadratic equation 9x2 - 3x + (m - 4) = 0 are such that α - β = 1, then the value of m is:1. 22. 13. 34. 4

Answer» Correct Answer - Option 1 : 2

Concept:

If α and β are the two roots of the quadratic polynomial Ax² + Bx + C, then:

Calculation:

Comparing the given equation 9x2 - 3x + (m - 4) = 0 with the general equation Ax2 + Bx + C = 0, we can say that:

A = 9, B = -3, C = (m - 4)

It is given that:

α - β = 1 ... (1)

Using α + β = \(\rm -\frac{B}{A}\), we get:

α + β = \(\rm -\left(\frac{-3}{9}\right)=\frac13\) ... (2)

Adding equations (1) and (2), we get:

\(\rm 2\alpha = 1+\frac13=\frac43\)

⇒ \(\rm \alpha = \frac23\)

Using equation (2), we get:

\(\rm \beta = \frac13-\frac23=-\frac13\)

Now, the product of the roots is: αβ = \(\rm \frac{C}{A}\).

⇒ \(\rm \left(\frac23\right)\times\left(-\frac13\right)=\frac{m-4}{9}\)

⇒ m - 4 = -2

⇒ m = 2.