If α and β are the zeroes of 3x² + 5x − 2, find / + / .

-37/6. / + / = ^2 + ^2/ . Here α + β = -5/3 and αβ = -2/3, so α² + β² = 25/9 + 4/3 = 37/9 and the value is 37/9/-2/3 = -37/6. Check: 3x² + 5x − 2 = (3x − 1)(x + 2), zeroes 1/3 and −2, giving -1/6 - 6 = -37/6 ✓. Using (α + β)² alone as α² + β² gives -25/6; dropping the minus on αβ gives 37/6.

If α and β are the zeroes of $3 x^{2} + 5 x - 2$ , find $\frac{α}{β} + \frac{β}{α}$ .

Class 10PolynomialsAlgebra — Polynomials

A $\frac{37}{6}$
B $- \frac{37}{9}$
C $- \frac{25}{6}$
D $- \frac{37}{6}$

Step-by-Step Solution

\frac{α}{β} + \frac{β}{α} = \frac{α ^{2} + β ^{2}}{α β}

. Here α + β =

- \frac{5}{3}

and αβ =

- \frac{2}{3}

, so α² + β² =

\frac{25}{9} + \frac{4}{3} = \frac{37}{9}

and the value is

\frac{37/9}{- 2/3} = - \frac{37}{6}

. Check:

3 x^{2} + 5 x - 2 = (3 x - 1) (x + 2)

, zeroes

\frac{1}{3}

and −2, giving

- \frac{1}{6} - 6 = - \frac{37}{6}

✓. Using

(α + β)^{2}

alone as α² + β² gives

- \frac{25}{6}

; dropping the minus on αβ gives

\frac{37}{6}

Answer: D)

- \frac{37}{6}

Key idea: Geometric meaning of zeros

Zeros of

p (x)

are exactly the **

x

-coordinates where the graph

y = p (x)

meets the

x

-axis**.

- **Linear**

y

= ax +

b

: graph is a straight line, crosses

x

-axis ONCE → exactly 1 zero.
- **Quadratic**

y = a x^{2} + b x + c

: graph is a parabola. Possible cases:
- Crosses

x

-axis at 2 points → 2 distinct real zeros.
- Touches

x

-axis at 1 point → 1 repeated zero (vertex on

x

-axis).
- Does not meet

x

-axis → no real zeros.
- **Cubic**

y = a x^{3}

+ …: shape is a curve that crosses 1, 2, or 3 times → 1, 2, or 3 real zeros.

So 'find the zeros' is the algebraic version of 'find where the graph crosses the

x

-axis'.

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If α and β are the zeroes of 3x2+5x−2, find βα​+αβ​.

Step-by-Step Solution

Key idea: Geometric meaning of zeros

Master this chapter in 30 minutes a day.

More Class 10 Polynomials practice

If α and β are the zeroes of $3 x^{2} + 5 x - 2$ , find $\frac{α}{β} + \frac{β}{α}$ .