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Kisachek [45]
2 years ago
6

Cedric completed the square for the quadratic expression 5x^2−40x+15 in order to determine the minimum value of the expression,

as shown.
Step 1: 5(x^2−8x+3)
Step 2: 5(x^2−8x+16−16+3)
Step 3: 5((x+4)^2−16+3)
Step 4: 5((x+4)^2−13)
Step 5: 5(x+4)^2−65
What mistake, if any, did Cedric make?
Mathematics
1 answer:
Olegator [25]2 years ago
5 0

Answer:

Step 3 is not correct (should be (x-4)^2, not (x+4)^2

Step-by-step explanation:

The expression that we have to simplify is:

5x^2-40x+15

We proceed step-by-step, in order to find the mistake did by Cedric:

1) First, we factorize the 5 outside:

5(\frac{5x^2}{5}-\frac{40x}{5}+\frac{15}{5})=5(x^2-40x+3) --> this step is correct

2) We add +16 and -16 inside the brackets:

5(x^2-8x+16-16+3) --> this step is correct

3) We rewrite the term (x^2-8x+16) as the square of a bynomial, which is (x-4)^2, so the expression becomes

5((x-4)^2-16+3) --> we notice that this step is wrong: in fact, Cedric wrote (x+4)^2, which is not correct.

4) Now we continue: we rewrite -16+3 as -13,

5((x-4)^2-13)

5) Finally, we multiply the 5 by the terms in the bracket:

=5(x-4)^2-65

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lapo4ka [179]

Answer: The correct options are  (1) (5,10), (2) (3,-3), (3) x = -1, (4) y=(x+2)^2+3, (5) 21s and (6) 0, -1, and 5.

Explanation:

Te standard form of the parabola is,

f(x)=a(x-h)^2+k        .....(1)

Where,  (h,k) is the vertex of the parabola.

(1)

The given equation is,

f(x)=(x-5)^2+10

Comparing this equation with equation (1),we get,

h=5 and k=10

Therefore, the vertex of the graph is (5,10) and the fourth option is correct.

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The given equation is,

f(x)=3x^2-18x+24

f(x)=3(x^2-6x)+24

To make perfect square add (\frac{b}{2a})^2, i.e., 9. Since there is factor 3 outside the parentheses, so subtract three times of 9.

f(x)=3(x^2-6x+9)+24-3\times 9

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Comparing this equation with equation (1),we get,

h=3 and k=-3

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(3)

The given equation is

f(x)=4x^2+8x+7

f(x)=4(x^2+2x)+7

To make perfect square add (\frac{b}{2a})^2, i.e., 1. Since there is factor 4 outside the parentheses, so subtract three times of 1.

f(x)=4(x^2+2x+1)+7-4

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Comparing this equation with equation (1),we get,

h=-1 and k=3

The vertex of the equation is (-1,3) so the axis is x=-1 and the correct option is 2.

(4)

The given equation is,

y=x^2+4x+7

To make perfect square add (\frac{b}{2a})^2, i.e., 2^2.

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Therefore, the correct option is  4.

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h=-16t^2+672t

It can be written as,

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It is a downward parabola. so the maximum height of the function is on its vertex.

The x coordinate of the vertex is,

x=\frac{b}{2a}

x=\frac{42}{2}

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(6)

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Use factoring method to find the factors of the equation.

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f(x)=3x(x(x-5)+1(x-5))

f(x)=3x(x-5)(x+1)

Equate each factor equal to 0.

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