Question

absolute value and quadratic functions and equations
unit 2 test: a

1. the area of the rectangle in the figure is 60 square meters. what are the width and length

of the rectangle? give your answer as an exact value. (simplified radical).
length:
width:

1. in the figure, the area of the larger square is 88 square inches greater than the area of the

smaller square. what are the lengths of the sides of each square? give your answer as an
exact value. (simplified radical).
length of small square side:
length of large square side:

1. determine if each quadratic equation has real solutions. choose yes or no.

a.) $\frac{2}{3}x^{2}+7 = 11$
b.) $12x^{2}-5=-3$
____ yes __ no __ yes ____ no

Explanation:

Step1: Solve for the rectangle (Problem 1)

Use the area formula \(A = lw\). Here \(l = 4x\), \(w=x\), and \(A = 60\). So \(4x\times x=60\), which simplifies to \(4x^{2}=60\). Then \(x^{2}=15\), and \(x=\sqrt{15}\) (since length cannot be negative). The length \(l = 4\sqrt{15}\) meters and the width \(w=\sqrt{15}\) meters.

Step2: Solve for the squares (Problem 2)

The area of the larger square is \((3x)^{2}\) and the area of the smaller square is \(x^{2}\). Given \((3x)^{2}-x^{2}=88\). Expand: \(9x^{2}-x^{2}=88\), so \(8x^{2}=88\), then \(x^{2}=11\), and \(x = \sqrt{11}\) (since side - length cannot be negative). The side - length of the small square is \(\sqrt{11}\) inches and the side - length of the large square is \(3\sqrt{11}\) inches.

Step3: Determine real solutions for quadratic equations (Problem 3)

• For \(A\):

Start with \(\frac{2}{3}x^{2}+7 = 11\). Subtract 7 from both sides: \(\frac{2}{3}x^{2}=4\). Multiply both sides by \(\frac{3}{2}\): \(x^{2}=6\). Since \(x=\pm\sqrt{6}\), there are real solutions.

• For \(B\):

Start with \(12x^{2}-5=-3\). Add 5 to both sides: \(12x^{2}=2\). Then \(x^{2}=\frac{2}{12}=\frac{1}{6}\). Since \(x = \pm\frac{1}{\sqrt{6}}=\pm\frac{\sqrt{6}}{6}\), there are real solutions.

Answer:

1. Length: \(4\sqrt{15}\) meters, Width: \(\sqrt{15}\) meters
2. Length of Small Square Side: \(\sqrt{11}\) inches, Length of Large Square Side: \(3\sqrt{11}\) inches
3. A. Yes, B. Yes