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astra-53 [7]
3 years ago
6

2. Write a quadratic equation with roots (-2,0) and (4,0) that has a minimum value of -36

Mathematics
1 answer:
Brums [2.3K]3 years ago
6 0

Answer:

f(x) = 4*(x+2)*(x-4) factorized formula

f(x) = 4x^2 - 8x - 32 polynomical formula

Step-by-step explanation:

Lets assume the factorized formula of a quadratic function:

f(x) = a*(x-x1)^2 * (x-x2)^2

where "a" is a coefficient and x1, x2 are the roots of the function. Then replacing the given roots:

f(x) = a*(x+2)*(x-4)

because its told us that -36 is the minimum value of the function we can say that its concave, then this value is actually the component in the Y axis of the vertex. To find the component in the X axis of the vertex we have to make the adding between the roots and divide this value by 2, this last is because a quadratic function is a symetrical function. Lets call the component at the X axis of the vertex as Xv, then:

Xv=(x1+x2)/2

Xv=(-2+4)/2

Xv=1

Therefore now we have a point of the function and its P=(1,-36) this point is the vertex of the function too.

Now the last thing to do is to find the value of the coefficient "a". We can find it by replacing the point of the vertex obtained before.

-36 = a*(1+2)*(1-4)

-36 = a*(3)*(-3)

-36 = a*(-9)

4 = a

Finally the equation is:

f(x) = 4*(x+2)*(x-4)

if we expand this function we find the polynomical form of this function

f(x) = 4x^2 - 8x - 32

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Line Segment DE is parallel to side BC of right triangle ABC. CD = 3, DE = 6, and EB = 4. Compute the area of quadrilateral BCDE
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The area of quadrilateral BCDE = 20.4 sq. units

Let AD = x and AE = y.

Since ΔABC and ΔAED are similar right angled triangles, we have that

AC/AD = AB/AE

AC = AD + CD

= x + 3.

Also, AB = AE + EB

= y + 4

So, AC/AD = AB/AE

(x + 3)/x = (y + 4)/y

Cross-multiplying, we have

y(x + 3) = x(y + 4)

Expanding the brackets, we have

xy + 3y = xy + 4x

3y = 4x

y = 4x/3

In ΔAED, AD² + AE² = DE².

So, x² + y² = 6²

Substituting y = 4x/3 into the equation, we have

x² + y² = 6²

x² + (4x/3)² = 6²

x² + 16x²/9 = 36

(9x² + 16x²)/9 = 36

25x²/9 = 36

Multiplying both sides by 9/25, we have

x² = 36 × 9/25

Taking square root of both sides, we have

x = √(36 × 9/25)

x = 6 × 3/5

x = 18/5

x = 3.6

Since y = 4x/3,

Substituting x into the equation, we have

y = 4 × 3.6/3

y = 4.8

To find the area of quadrilateral BCDE, we subtract the area of ΔAED from area of ΔABC.

So, area of quadrilateral BCDE = area of ΔABC - area of ΔAED

area of ΔABC = 1/2 AC × AB

= 1/2 (x + 3)(y + 4)

= 1/2(3.6 + 3)(4.8 + 4)

= 1/2 × (6.6)(8.8)

= 1/2 × 58.08

= 29.04  square units

area of ΔAED = 1/2 AD × AE

= 1/2xy

= 1/2 × 3.6 × 4.8

= 1/2 × 17.28

= 8.64 square units

area of quadrilateral BCDE = area of ΔABC - area of ΔAED

area of quadrilateral BCDE = 29.04 sq units - 8.64 sq units

area of quadrilateral BCDE = 20.4 sq. units

So, the area of quadrilateral BCDE = 20.4 sq. units

Learn more about area of a quadrilateral here:

brainly.com/question/19678935

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