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Help with the following questions, 1. Find all solutions to the equation $x^2 - 7x = 30$. If you find more than one, then list the values separated by commas. If the solutions are not real, then they should be written in $a + bi$ form. 2.Find all solutions to the equation $6x^2 - 31x + 40 = 0.$ If you find more than one, then list the values separated by commas. If the solutions are not real, then they should be written in $a + bi$ form. 3.The quadratic equation $2x^2 + bx + 18 = 0$ has a double root. Find all possible values of $b.$ 4.Let $r$ and $s$ be the roots of $x^2 - 6x + 2 = 0.$ Find $(r - s)^2.$ 5.Find all solutions to the equation $x^2 + 29 = 10x$. 6.Find all values of $c$ such that $\dfrac{c}{c-5} = \dfrac{4}{c-4}$.

 Jan 19, 2021
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Q1. 

Rearranging the equation,

\(x^2 - 7x - 30 = 0\)

Then, using the quadratic formula, 

\(x = \dfrac{7 \pm \sqrt{7^2 - 4\cdot 1 \cdot (-30)}}{2}\)

I will leave the simplification for you.

 

Q2. Similar to Q1. If you understand what I did in Q1, then it's not hard. Just repeat what I did in Q1.

 

Q3. Keyword: "the ... equation ... has a double root."

The discriminant of a quadratic equation which has a double root is 0.

 

\(b^2 - 4\cdot 2 \cdot 18 = 0\\ b^2 - 12^2 = 0\\ (b - 12)(b + 12) = 0\)

Then this means b - 12 = 0 or b + 12 = 0. This is just one step from the answer, I will let you figure out this part.

 

Q4. This is a root relation problem. We will use the formulae for relations of roots.

 

\(\begin{cases} r + s = -\dfrac{(-6)}{1} = 6\\ rs = \dfrac{2}1 = 2 \end{cases}\)

 

We then express (r - s)^2 in terms of r + s and rs. 

 

\((r - s)^2 = r^2 - 2rs + s^2 = (r^2 + 2rs + s^2) - 4rs = (r + s)^2 -4rs\)

 

You can substitute the values of r + s and rs directly.

 

Q5. Similar to Q1. Repeat what I did.

 

Q6. 

\(\dfrac c{c - 5} = \dfrac4{c - 4}\)

 

Cross-multiplying,

\(c(c - 4) = 4(c- 5)\)

 

Expanding,

\(c^2 - 4c = 4c - 20\)

 

Moving everything to the left-hand side,

\(c^2 - 8c + 20 = 0\)

 

Then repeat what I did in Q1.

 

P.S. The entire "Quadratic Equation" part of the syllabus is for solving equations of the form ax^2 + bx + c = 0, so make sure you put the question in this form before you proceed.

 

P.P.S. Some parts need knowledge of discriminant \(\Delta = b^2 - 4ac\) and the root relation formulae. 

 

P.P.P.S. If you don't understand any part of my solutions, feel free to ask me.

 Jan 19, 2021

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