Explanation

For 2x² - 3x + 1 = 0:  

Sum = 3/2, Product = 1/2  

Let roots be α, β. Then new roots are 1/α, 1/β  

New sum = 1/α + 1/β = (α + β)/αβ = (3/2) / (1/2) = 3  

New product = 1/αβ = 1/(1/2) = 2  

New equation: x² - (sum)x + product = x² - 3x + 2 = 0  

So a:b:c = 1:-3:2

Last verified on 06-05-2026

Explanation

(x + m)(x + n) = x² + (m + n)x + mn  

So p = m + n = 10  

q = mn = 21  

p + q = 10 + 21 = 31  

Last verified on 06-05-2026

Explanation

(a + bi)² = a² - b² + 2abi  

Purely real means imaginary part = 0  

So 2ab = 0  

Given a ≠ 0, then b = 0

Last verified on 06-05-2026

Explanation

Explanation

Make b the same in both ratios  

a:b = 3:4  

b:c = 2:5 → multiply by 2 → b:c = 4:10  

Now a:b:c = 3:4:10  

So a:c = 3:10

Last verified on 04-05-2026

Explanation

• In geometry, a tangent is a line that touches a circle at only one point.
• At the point of contact, it forms a right angle (90°).
• The tangent is always perpendicular to the radius of the circle.

Last verified on 04-05-2026

Explanation

Identity: 1 + tan²θ = sec²θ

Last verified on 04-05-2026

Explanation

For similar triangles: area ratio = (side ratio)²  

(side ratio)² = 9/16  

side ratio = √9/16

side ratio = 3/4

Last verified on 04-05-2026

Explanation

Right triangle: a² + b² = c²  

5² + b² = 13²  

25 + b² = 169  

b² = 169 - 25  

b² = 144  

b = 12  

Last verified on 04-05-2026

Explanation

Third side rule: difference < x < sum  

10 - 7 < x < 10 + 7  

3 < x < 17

16 → 3 < 16 < 17, possible

Last verified on 04-05-2026