Find the value of x. AB = 4x - 15 ,BC = 2x + 3 and AC (hypotenuse)= 48.

2 Answers | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

The problem provides the information that AB,BC are the legs of right triangle and AC represents the hypotenuse, hence, you need to use Pythagora's theorem, such that:

`AC^2 = AB^2 + BC^2`

Replacing 48 for AC, `4x - 15` for AB and `2x + 3` for BC yields:

`48^2 = (4x - 15)^2 + (2x + 3)^2`

`48^2 - (2x + 3)^2 = (4x - 15)^2 `

`(48 - (2x + 3))(48 + 2x + 3) = 16x^2 - 120x + 225`

`(45 - 2x)(51 + 2x) = 16x^2 - 120x + 225`

`2295 + 90x - 102x - 4x^2 = 16x^2 - 120x + 225`

`20x^2 - 108x - 2070 = 0 => 10x^2 - 54x - 1035 = 0`

You ened to complete the square `10x^2 - 54x` such that:

`a^2 = 10x^2 => a = xsqrt10`

`2ab = 54x => 2*x*sqrt10*b = 54 x`

`b = 27/sqrt10`

`10x^2 - 54x + 729/10 = 1035 + 729/10`

`(x*sqrt10 - 27/sqrt10)^2 = 11079/10`

`x*sqrt10 - 27/sqrt10 = +-sqrt(11079/10)`

`x*sqrt10 = (27+-sqrt11079)/sqrt10 => x = (27+-sqrt11079)/10`

You need to exclude `x = (27-sqrt11079)/10` as it creates a negative length of a leg.

Hence, evaluating x, under the given conditions, yields `x = (27+sqrt11079)/10.`

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Since the given sides represent the cathetus and the hypothenuse of a right angle triangle, we'll determine x applying Pythagorean theorem:

hypothenuse^2 = cathetus^2 + cathetus^2

From enunciation, the hypothenuse is AC:

AC^2 = AB^2 + BC^2

48^2 = (4x - 15)^2 + (2x + 3)^2

We'll expand the square form the right side:

2304 = 16x^2 - 120x + 225 + 4x^2 + 12x + 9

We'll combine like terms and we'll use symmetric property:

20x^2 - 108x - 2070 = 0

10x^2 - 54x - 1035 = 0

We'll apply the quadratic formula:

x1 = [54+sqrt(44316 )]/20

x1 = (54+210.51)/20

x1 = 13.2255 approx.

x2 = -7.8255 approx.

We’ve answered 318,915 questions. We can answer yours, too.

Ask a question