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Answer:
Let the two positive number be x and y.
Assume [tex]x>y[/tex]
From the given condition: The sum of 2 positive number is 151 and the lesser number is 19 more than the square root of the greater number.
then we have,
[tex]x+y=151[/tex] .....[1]
[tex]y= 19+\sqrt{x}[/tex]
Substitute the value of [tex]y= 19+\sqrt{x}[/tex] in equation [1],
[tex]x+19+\sqrt{x} =151[/tex]
Subtract 19 from both the sides, we get
[tex]x+\sqrt{x}+19-19=151-19[/tex]
on simplify:
[tex]x+\sqrt{x} = 132[/tex] or
[tex]\sqrt{x} =132-x[/tex]
squaring both the sides, we get
[tex](\sqrt{x} )^2=(132-x)^2[/tex]
Using Identities [tex](a-b)^2=a^2+b^2-2ab[/tex] on right hand side of above expression:
[tex]x=132^2 +x^2-2\cdot 132\cdot x[/tex] or
[tex]x= 17424 +x^2-264x[/tex] or we can write it as:
[tex]x^2-265x+17424 =0[/tex]
By, solving above quadratic equation we get,
x =121
to find the value of y:
[tex]x+y=151[/tex]
Substitute the value of x in above equation to get value for y:
[tex]121+y=151[/tex]
⇒ y= 30
The value of greatest number minus the lesser would be x-y, i.e, 121-30 = 91
The value of the greater number minus the lesser number is [tex]\boxed{191}[/tex].
Further Explanation:
Let the greater positive number be [tex]\boxed{x}[/tex].
Let the lesser positive number be [tex]\boxed{y}[/tex].
The first condition is that the sum of [tex]2[/tex] positive numbers is [tex]151[/tex].
The equation from the first condition can be expressed as,
[tex]\boxed{x + y = 151}[/tex]
The second condition is that the lesser number is [tex]19[/tex] more than the square root of the greater number.
[tex]\boxed{y = 19 + \sqrt x }[/tex]
Substitute the value [tex]y = 19 + \sqrt x[/tex] in equation [tex]x + y = 151[/tex] to obtain the value of [tex]x[/tex].
[tex]\begin{aligned} x +\left( {19+\sqrt x }\right)&= 151 \\ x + \sqrt x &= 151 - 19 \\ x + \sqrt x &= 132 \\ \sqrt x &= 132 - x \\ \end{aligned}[/tex]
Square both the sides.
[tex]\boxed{{\left( {\sqrt x } \right)^2} = {\left( {132 - x} \right)^2}}[/tex]
Use the identity [tex]\boxed{{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab}[/tex].
[tex]\begin{aligned} x &= {132^2} + {x^2} - 2 \times 132 \times x \\ x &= {132^2} + {x^2} - 264x \\ 0 &= {x^2} - 265x + 17424 \\\end{aligned}[/tex]
Solve the above quadratic equation to obtain the value of [tex]x[/tex].
[tex]\begin{aligned} {x^2} - 121x - 144x + 17424 &= 0 \\ x\left( {x - 121} \right) - 144\left( {x - 121} \right) &= 0 \\ \left( {x - 121} \right) \times \left( {x - 144} \right) &= 0 \\\end{aligned}[/tex]
The value of [tex]x[/tex] can be [tex]121[/tex] and [tex]144[/tex].
The value of [tex]x=144[/tex] cannot be possible as it doesn’t satisfies the two condition. Therefore the value of [tex]x[/tex] is [tex]\boxed{121}[/tex].
The greater number is [tex]\boxed{121}[/tex].
Substitute [tex]121[/tex] for [tex]x[/tex] in equation [tex]x + y = 151[/tex] to obtain the value of [tex]y[/tex].
[tex]\begin{aligned}121 + y &= 151 \\y &= 151 - 121 \\ y &= 30 \\\end{aligned}[/tex]
The difference between the greater number and smaller number can be obtained as,
[tex]\begin{aligned}\\\text{Difference} &= x - y \\ &= 121 - 30 \\ &= 91 \\\end{aligned}[/tex]
The value of the greater number minus the lesser number is [tex]\boxed{191}[/tex].
Learn more:
1. Learn more about unit conversion https://brainly.com/question/4837736
2. Learn more about non-collinear https://brainly.com/question/4165000
Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Quadratic Equations
Keywords: sum, positive number, square root, lesser number, greater, number, quadratic equation, minus, sum of 2 positive numbers is 151, difference, lesser number is 19 more than the square root of greater, number, value.
