Question

spiral: keeping it fresh!
**#5.) line p ⊥ line m and has the equation 4x − 6y = 18. given that line m intersects line p at its y-intercept, write the equation of line m and graph and label the line below. grid image
*#6.) solve for the roots of the equation 39 = −36x + 3x²
#7.) simplify the expression (3x − 2)².
a. 6x − 4
b. 9x² − 12x + 4
c. 9x² − 4
d. 9x² + 12x − 4
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Explanation:

Problem #5

Step1: Find slope of line p

Rewrite \(4x - 6y = 18\) in slope - intercept form \(y=mx + b\) (where \(m\) is slope, \(b\) is y - intercept).
Subtract \(4x\) from both sides: \(-6y=-4x + 18\).
Divide by \(-6\): \(y=\frac{4}{6}x-3=\frac{2}{3}x - 3\). So slope of line \(p\), \(m_p=\frac{2}{3}\).

Step2: Find slope of line m

Since line \(p\perp\) line \(m\), the product of their slopes is \(- 1\) (\(m_p\times m_m=-1\)).
Let \(m_m\) be slope of line \(m\). Then \(\frac{2}{3}\times m_m=-1\), so \(m_m=-\frac{3}{2}\).

Step3: Find y - intercept of line p (and line m)

From \(y = \frac{2}{3}x-3\), y - intercept of line \(p\) is \(b=-3\). Since line \(m\) intersects line \(p\) at its y - intercept, line \(m\) also passes through \((0, - 3)\), so its y - intercept \(b=-3\).

Step4: Write equation of line m

Using slope - intercept form \(y = m_mx + b\), with \(m_m=-\frac{3}{2}\) and \(b = - 3\), we get \(y=-\frac{3}{2}x-3\).

Step1: Rewrite the equation in standard form

Given \(39=-36x + 3x^{2}\), rewrite as \(3x^{2}-36x - 39 = 0\).
Divide all terms by 3: \(x^{2}-12x - 13 = 0\).

Step2: Factor the quadratic equation

We need two numbers that multiply to \(-13\) and add to \(-12\). The numbers are \(-13\) and \(1\).
So, \(x^{2}-12x - 13=(x - 13)(x+1)=0\).

Step3: Solve for x

Set each factor equal to zero:
\(x - 13 = 0\) gives \(x = 13\);
\(x + 1=0\) gives \(x=-1\).

Step1: Use the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\)

For \((3x-2)^{2}\), let \(a = 3x\) and \(b = 2\).

Step2: Substitute into the formula

\((3x)^{2}-2\times(3x)\times2+2^{2}=9x^{2}-12x + 4\).

Answer:

The equation of line \(m\) is \(y =-\frac{3}{2}x-3\) (For graphing: Plot the y - intercept \((0,-3)\), then use slope \(-\frac{3}{2}\) (down 3, right 2 or up 3, left 2) to find another point and draw the line, label it as line \(m\))

Problem #6