Question

The painting shown at the right has an area of 260 in^2. What is the value of x?

Answer 1

x is about 10.68 in

Explanation:

The formula for area of a rectangle is given by the formula:

A = lw, where

• A is the area in square units,
• l is the length,
• and w is the width

Step 1: We can plug in 260 for A, (2x + 3) for l, and x for w:

260 = (2x + 3)(x)

Step 2: Multiply x and (2x + 3)

260 = x * 2x + x * 3

260 = 2x^2 + 3x

Step 3: Subtract 260 from both sides to get a regular quadratic equation in standard form:

(260 = 2x^2 + 3x) - 260

0 = 2x^2 + 3x - 260

Currently 2x^2 + 3x - 260 is in standard form, whose general equation is given by:

0 = ax^2 + bx + c

We can solve for the roots of the quadratic equation using the quadratic formula, which is given by:

`x=(-b+/-√(b^2-4ac) )/(2a)`, where

• x is the roots (solution to quadratic equation)
• The +/- comes from that fact that taking the square root of a number gives us both a positive and negative answer

Since we know that 2 is our a value, 3 is our b value, and -260 is our c value, we plug these in for a, b, and c in the quadratic formula:

Positive answer:

`x=(-3+√(3^2-4(2)(-260)) )/(2(2))\\ x=(-3+√(9+2080) )/(4)\\ x=-3/4+1/4√(2089)\\ x=10.67639488\\x = 10.68`

If you were to find the negative answer, you'd get x is about -18.93. Because we can't have a negative answer, we must use the positive answer and thus x is about 10.68

Optional Step 3: We can check that x = 10.68 is the correct answer by plugging in 10.68 for x in the formula A = (2x + 3)(x) and check that we get 260 or something very close to it:

260 = (2 * 10.68 + 3)(10.68)

260 = (21.36 + 3)(10.68)

260 = (24.36)(10.68)

260 > 249.4848

Since we rounded x to the nearest hundredth, you get an approximate answer. You'd get an exact answer if you used the unrounded answer, which I did on my graphing calculator. I got 249.4848 when I used the rounded answer for x and 260 when I used the exact answer for x.