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Wittaler [7]
3 years ago
13

the product of 2 consecutive integers is 5 more than the square of the smaller integer.What is the value of the smaller integer?

Mathematics
1 answer:
Dvinal [7]3 years ago
4 0

Answer:

5

Step-by-step explanation:

lets say that the small number is x, the consecutive one must be x+1

x(x+1) = x²+5 ; distribute x

x²+x = x²+5 ; subtract x² from both sides

x= 5

check :

5*6 = 30

5²+5 = 25+5 = 30

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The exact value is found by making use of order of operations. The

functions can be resolved using the characteristics of quadratic functions.

Correct responses:

  • \displaystyle 1\frac{4}{7} \div \frac{2}{3} - 1\frac{5}{7} =\frac{9}{14}
  • x = -4, y = 12
  • When P = 1, V = 6
  • \displaystyle x = -3 \ or \ x = \frac{1}{2}

\displaystyle i. \hspace{0.1 cm} \underline{ f(x) = 2 \cdot \left(x - 1.25 \right)^2 + 4.875 }

ii. The function has a minimum point

iii. The value of <em>x</em> at the minimum point, is <u>1.25</u>

iv. The equation of the axis of symmetry is <u>x = 1.25</u>

<h3>Methods by which the above responses are found</h3>

First part:

The given expression, \displaystyle \mathbf{ 1\frac{4}{7} \div \frac{2}{3} -1\frac{5}{7}}, can be simplified using the algorithm for arithmetic operations as follows;

  • \displaystyle 1\frac{4}{7} \div \frac{2}{3} - 1\frac{5}{7} = \frac{11}{7}  \div \frac{2}{3} - \frac{12}{7} = \frac{11}{7} \times \frac{3}{2} - \frac{12}{7} = \frac{33 - 24}{14} =\underline{\frac{9}{14}}

Second part:

y = 8 - x

2·x² + x·y = -16

Therefore;

2·x² + x·(8 - x) = -16

2·x² + 8·x - x² + 16 = 0

x² + 8·x + 16 = 0

(x + 4)·(x + 4) = 0

  • <u>x = -4</u>

y = 8 - (-4) = 12

  • <u>y = 12</u>

Third part:

(i) P varies inversely as the square of <em>V</em>

Therefore;

\displaystyle P \propto \mathbf{\frac{1}{V^2}}

\displaystyle P = \frac{K}{V^2}

V = 3, when P = 4

Therefore;

\displaystyle 4 = \frac{K}{3^2}

K = 3² × 4 = 36

\displaystyle V = \sqrt{\frac{K}{P}

When P = 1, we have;

\displaystyle V =\sqrt{ \frac{36}{1} } = 6

  • When P = 1, V =<u> 6</u>

Fourth Part:

Required:

Solving for <em>x</em> in the equation; 2·x² + 5·x  - 3 = 0

Solution:

The equation can be simplified by rewriting the equation as follows;

2·x² + 5·x - 3 = 2·x² + 6·x - x - 3 = 0

2·x·(x + 3) - (x + 3) = 0

(x + 3)·(2·x - 1) = 0

  • \displaystyle \underline{x = -3 \ or\ x = \frac{1}{2}}

Fifth part:

The given function is; f(x) = 2·x² - 5·x + 8

i. Required; To write the function in the form a·(x + b)² + c

The vertex form of a quadratic equation is f(x) = a·(x - h)² + k, which is similar to the required form

Where;

(h, k) = The coordinate of the vertex

Therefore, the coordinates of the vertex of the quadratic equation is (b, c)

The x-coordinate of the vertex of a quadratic equation f(x) = a·x² + b·x + c,  is given as follows;

\displaystyle h = \mathbf{ \frac{-b}{2 \cdot a}}

Therefore, for the given equation, we have;

\displaystyle h = \frac{-(-5)}{2 \times 2} = \mathbf{ \frac{5}{4}} = 1.25

Therefore, at the vertex, we have;

k = \displaystyle f\left(1.25\right) = 2 \times \left(1.25\right)^2 - 5 \times 1.25  + 8 = \frac{39}{8} = 4.875

a = The leading coefficient = 2

b = -h

c = k

Which gives;

\displaystyle f(x) \ in \ the \ form \  a \cdot (x + b)^2 + c \ is \ f(x) = 2 \cdot \left(x + \left(-1.25 \right) \right)^2 +4.875

Therefore;

  • \displaystyle \underline{ f(x) = 2 \cdot \left(x -1.25\right)^2 + 4.875}

ii. The coefficient of the quadratic function is <em>2</em> which is positive, therefore;

  • <u>The function has a minimum point</u>.

iii. The value of <em>x</em> for which the minimum value occurs is -b = h which is therefore;

  • The x-coordinate of the vertex = h = -b =<u> 1.25 </u>

iv. The axis of symmetry is the vertical line that passes through the vertex.

Therefore;

  • The axis of symmetry is the line <u>x = 1.25</u>.

Learn more about quadratic functions here:

brainly.com/question/11631534

6 0
2 years ago
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