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Let side of bigger square is x m. And, side of smaller square is y m. According to given condition, x 2 + y 2 = 260 ...(i) And 4x – 4y = 24 ⇒ 4(x – y) = 24 ⇒ x – y = 6 ⇒ x = 6 + y ...(ii) Putting the value of (ii) in (i), we get (6 + y) 2 + y 2 = 260 ⇒ 36 + y 2 + 12y + y 2 = 260 ⇒ 2y 2 + 12y – 224 = 0 ⇒ y 2 + 6y – 112 = 0 ⇒ y 2 + 14y – 5y – 112 = 0 ⇒ y(y + 14) –8(y + 14) = 0 ⇒ (y – 8) (y + 14) = 0 ⇒ y–8 = 0, y + 14 = 0 ⇒ y = 8, y = – 14 Putting the value of y in (ii), we get x = 6 + y = 6 + 8 = 14. Hence, sides of squares are 14 cm and 8 cm.

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Quadratic Equations

Short Answer Type

161. Find the roots of the quadratic equation by using quadratic formula:

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162. Find the roots of the quadratic equation by using quadratic formula:

9 straight x squared plus 12 straight x plus 4 equals 0

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163. Find the roots of the quadratic equation by using quadratic formula:

straight x squared plus straight x minus 156 equals 0

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164. Find the roots of the quadratic equation by using quadratic formula:

space space space space straight y squared equals 41 straight y minus 400.

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165. Use quadratic formula to solve the following equation for x:
abx² + (b² – ac) x – bc = 0.

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166. Use quadratic formula to solve the following equation for x:
9x² – 9 (a + b)x + (2a² + 5ab + 2b²) = 0.

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167. Use quadratic formula to solve the following equation for x:
a²x² + (a² – b²)x – b² = 0.

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168. A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.

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Long Answer Type

169. Sum of the areas of two squares is 260 m². If the difference of their perimeters is 24 m, then find the sides of the two squares.

Let side of bigger square is x m. And, side of smaller square is y m.
According to given condition,
x² + y² = 260 ...(i)
And 4x – 4y = 24
⇒ 4(x – y) = 24
⇒ x – y = 6
⇒ x = 6 + y ...(ii)
Putting the value of (ii) in (i), we get
(6 + y) 2 + y² = 260
⇒ 36 + y² + 12y + y² = 260
⇒ 2y² + 12y – 224 = 0
⇒ y² + 6y – 112 = 0
⇒ y² + 14y – 5y – 112 = 0
⇒ y(y + 14) –8(y + 14) = 0
⇒ (y – 8) (y + 14) = 0
⇒ y–8 = 0, y + 14 = 0
⇒ y = 8, y = – 14
Putting the value of y in (ii), we get
x = 6 + y
= 6 + 8 = 14.
Hence, sides of squares are 14 cm and 8 cm.