Question

parts a,b,e,f, g

1. simplify the following

a. \\(sqrt{18}\\)
b. \\(sqrt{20a^3b^4}\\)
c. \\(sqrt{-64}\\)
e. \\(sqrt{16a^3b^5}\\)
f. \\(sqrt{8x^5y^2}\\)
g. \\(sqrt3{54x^6y^9z^3}\\)

1. find the root(s) of \\(y = -2x^2 + 6x + 1\\). leave your answer as a simplified radical, if necessary.

\\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\)

simplified radical form \\(x = \underline{\quad\quad}\\) and \\(x = \underline{\quad\quad}\\)
decimal form (round to the nearest tenth) \\(x = \underline{\quad\quad}\\) and \\(x = \underline{\quad\quad}\\)

1. simplify each algebraic expression.

a. \\((9x^2 + x - 5) + (13x^2 - 5x + 12)\\)
b. \\((-10x^2 + 4x + 5) - (5x^2 - 7x - 9)\\)

Explanation:

Step1: Identify the quadratic equation

The equation is \( y = -2x^2 + 6x + 1 \), so for the quadratic formula \( ax^2+bx+c = 0 \), we have \( a=-2 \), \( b = 6 \), \( c = 1 \).

Step2: Apply the quadratic formula

The quadratic formula is \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \). Substitute \( a=-2 \), \( b = 6 \), \( c = 1 \) into it:
First, calculate the discriminant \( D=b^2-4ac=(6)^2-4\times(-2)\times(1)=36 + 8=44 \).
Then, \( x=\frac{-6\pm\sqrt{44}}{2\times(-2)}=\frac{-6\pm2\sqrt{11}}{-4}=\frac{6\mp2\sqrt{11}}{4}=\frac{3\mp\sqrt{11}}{2} \).

Step3: Simplify the radical form

For \( x=\frac{3+\sqrt{11}}{2} \) and \( x=\frac{3 - \sqrt{11}}{2} \).

Step4: Calculate the decimal form

\( \sqrt{11}\approx3.3166 \)
For \( x=\frac{3+\sqrt{11}}{2}\approx\frac{3 + 3.3166}{2}=\frac{6.3166}{2}\approx3.2 \)
For \( x=\frac{3-\sqrt{11}}{2}\approx\frac{3-3.3166}{2}=\frac{-0.3166}{2}\approx - 0.2 \)

Answer:

Simplified Radical form \( x = \frac{3+\sqrt{11}}{2} \) and \( x=\frac{3 - \sqrt{11}}{2} \)
Decimal Form \( x\approx3.2 \) and \( x\approx - 0.2 \)