CRE 3 - Forming a Quadratic Equation and Relation between roots and coefficients

We know a quadratic equation whose roots are α and β can be written as (x - α)(x - β) = 0.

∴ The required quadratic equation is (x - 5)(x – 6) = 0

⇒ x² – 5x – 6x + 30 = 0

⇒ x² – 11x + 30 = 0

Hence, option (b).

We know a quadratic equation can be written as x² – (sum of the roots)x + (product of the roots) = 0

∴ The required quadratic equation is:

⇒ x² – 2x – 15 = 0

Hence, option (b).

Given that α and β are the roots of the equation 2x² - 3x - 5 = 0.

∴ α + β = $-\frac{-3}{2}$ = $\frac{3}{2}$ and α × β = ${\frac{-5}{2}}_{}$

Now, $\frac{1}{\alpha }$+ $\frac{1}{\beta }$ = $\frac{\alpha +\beta }{\alpha \beta }$ = $\frac{3/2}{-5/2}$ = $-\frac{3}{5}$

Hence, option (a).

We know, (α + ß)³ = α³ + ß³ + 3αß(α + ß)

⇒ α³ + ß³ = (α + ß)³ - 3αß(α + ß)   …(1)

Here, α + ß = sum of the roots = - (-3/1) = 3

and, αß = product of the roots = 5/1 = 5.

Substituting these values in (1), we get

α³ + ß³ = (3)³ – 3 × 5 × (3)

⇒ α³ + ß³ = 27 – 45 = -18

Hence, option (b).

The roots of the equatino 9x² - 36x + a = 0 are in the ratio 2 : 1.

Let the roots be α and 2α.

⇒ Sum of the roots = -b/a
⇒ α + 2α = -(-36)/9 = 4
⇒ 3α = 4
⇒ α = 4/3

Now, product of the roots = c/a
⇒ α × 2α = a/9
⇒ 18α² = a
⇒ a = 18 × 16/9 = 32.

Hence, option (b).

We know, in a quadratic equation ax² + bx + c = 0,

Sum of the roots = -b/a and product of the roots = c/a

Lomror wrote the contant term wrong but he wrote the coefficient of x correctly, hence the sum of the roots he got would be same as the sum of the roots of the actual equation.

∴ Sum of the roots of the actual equation = 4 + 3 = 7

Rohit wrote the coefficient of x wrong but he wrote the constsant term correctly, hence the product of the roots he got would be same as the product of the roots of the actual equation.

∴ Product of the roots of the actual equation = 2 × 3 = 6

Hence, the original equation is x² - 7x + 6 = 0.

⇒ (x - 6)(x - 1) = 0

⇒ x = 1 or 6.

Hence, option (c).

We know, in a quadratic equation with rational coefficients if one root is a + √b, the other root will be a - √b.

Hence, if one the roots of the given quadratic equation is 3 + 2√2, the other root will be 3 - 2√2.

∴ Sum of the roots = 3 + 2√2 + 3 - 2√2 = 6, and

Product of the roots = (3 + 2√2) × (3 - 2√2) = 9 - 8 = 1

∴ The quadratic equation is x² - (Sum)x + Product = 0

⇒ x² - 6x + 1 = 0

Hence, option (c).