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Triangle ABC is right angled at A. The area of triangle is = ½bh. The area of the triangle is 3/2x^2 + 10x + 16cm^2 where x is a positive integer.
The area of the triangle ABC can be written in the form 1/2 (x+a)(bx + c) . The value of a+b+c is____________.
I don't understanding, PLease kindly help me solve this?
3/2 x^2 + 10x + 16 = 1/2 (x+4)(3x+8)
This is just an exercise in factoring. Now you can easily read off a,b,c
Thank you!
To solve this problem, we need to find the values of a, b, and c in the given expression 1/2 (x+a)(bx + c) that represents the area of triangle ABC.
We're given that the area of the triangle is equal to 3/2x^2 + 10x + 16 cm^2. We want to write this expression in the form (x+a)(bx + c).
First, let's expand (x+a)(bx + c) to get an expression with three terms:
(x+a)(bx + c) = x(bx + c) + a(bx + c) = b(x^2) + cx + abx + ac.
Now, we can equate this expanded form with the given area of the triangle:
b(x^2) + cx + abx + ac = 3/2x^2 + 10x + 16.
To find the values of a, b, and c, we need to match the coefficients of each term on both sides of the equation.
Comparing the coefficients of the x^2 term, we have:
b = 3/2.
Comparing the coefficients of the x term, we have:
c + ab = 10.
Comparing the constant terms, we have:
ac = 16.
Now, we have a system of equations to solve for a, b, and c.
From the first equation, we can solve for b:
b = 3/2.
Next, from the third equation, we can solve for a:
a = 16/c.
Substituting this value of a in the second equation, we get:
c + (16/c)(3/2) = 10.
Multiplying both sides by 2c:
2c^2 + 48/2 = 10c.
2c^2 + 24 = 10c.
Rearranging the equation:
2c^2 - 10c + 24 = 0.
Now, we can solve this quadratic equation using factoring or the quadratic formula.
Let's use the quadratic formula:
c = (-b ± sqrt(b^2 - 4ac)) / 2a.
Plugging in the values:
c = (-(-10) ± sqrt((-10)^2 - 4(2)(24))) / (2(2)).
Simplifying:
c = (10 ± sqrt(100 - 192)) / 4.
c = (10 ± sqrt(-92)) / 4.
Since we need a positive integer value for x, we can see that the value inside the square root is negative. Therefore, there are no real solutions for c in this case.
However, we can still find the values of a and b using the equations we obtained earlier:
b = 3/2.
a = 16/c = 16/0. This is undefined.
Therefore, there is no solution for the values of a, b, and c that would allow us to write the expression 1/2(x+a)(bx + c) in the given form.