Master Algebra 1 Word Problems with our Comprehensive Practice

Simplify $$8-<!--\; -\; -->8^2+8÷<!--\; ÷\; -->7$$.

What is the product of 104 and -224?

What is 671 over 11?

What is -957 over -29?

What is one hundred two plus two hundred seventy-six?

How to prove that if a quartic equation ( with real coefficients ) has 4 imaginary roots they all will be in conjugate pairs?
I proved this fact for qudratic equation in the following way , let a qudratic equation have a imaginary root ${\displaystyle p+iq}$(q is not 0), and let other root be ${\displaystyle (a+ib)}$. Now here sum of roots will be a real number lets say R, ${\displaystyle \Rightarrow (p+iq)(a-iq)=R^{\prime }\Rightarrow (pa+q^2)+i(aq-pq)=0\Rightarrow aq-pq=0\Rightarrow aq=pq\Rightarrow a=p}$. So finally the roots are, ${\displaystyle p+iq\text{and}p-iq}$, hence proved. I tried to prove this fact for quartic equation in a similar way but could not reach to the conclusion. Please guide me by answering by my method or by suggesting any other simple method to prove that if a quartic equation has all imaginary roots then they will occur in conjugate pairs.

Finding the x values of a quadratic that has many different variables
The two solutions of the quadratic equation ${\displaystyle 3x^2-x+k=0\text{are}\frac{p}{4}\text{and}p+1}$. Determine the values of k and p.

Finding the value of a variable in a quadratic
The number of negative integral values of m for which the expression ${\displaystyle x^2+2(m-1)x+m+5}$ is positive ${\displaystyle \mathrm{\forall }x>1}$ is?

Finding ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{f\left(n\right)}}$ where f is a real quadratic function?

Finding integer a, b such that the roots of
${\displaystyle 3x^2+3(a+b)x+4ab=0}$
satisfy
${\displaystyle \alpha (\alpha +1)+\beta (\beta +1)=(\alpha +1)(\beta +1)}$

Finding extremum of quadratic function using factorisation as symmetry operation.

Finding a quadratic equation given its two roots. Does this method exist?
Quadratic equation given roots:
${\displaystyle 1-2\sqrt{3},1+2\sqrt{3}}$

Find the z-transform for the sequences
1) ${\displaystyle x\left(n\right)=10u\left(n\right)}$
2) ${\displaystyle x\left(n\right)=10\sin \left(0.25\pi n\right)u\left(n\right)}$

Find the Z-transform for the sequence ${\displaystyle x\left[n\right]=2^{\nu }[-n]+{\left(\frac{1}{4}\right)}^{\nu }[n-1]}$

Find the sum of the infinite geometric sequences: ${\displaystyle 48,24,12\dots }$

Find the quadratic equation from given relatioship
The quadratic equation whose roots are a and b where
${\displaystyle a^2+b^2=5\text{and}3(a^5+b^5)=11(a^3+b^3)}$

Find all integers ${\displaystyle n\left(n>0\right)}$ such that the which quadratic equation
${\displaystyle a_{n+1}{x}^2-2x\sqrt{\sum _{k=1}^{n+1}{a}_k^2}+\sum _{k=1}^n{a}_k=0}$
has real roots for every choice of real numbers ${\displaystyle a_1,a_2,\cdots ,a_{n+1}}$.

Figuring out coefficients of composition of a first degree polynomial into a quadratic
Given: ${\displaystyle g\left(x\right)=x^2+x-2}$ and ${\displaystyle g\circ f=2[2x^2-5x+2]}$, find f(x)(f(x) is form ${\displaystyle ax+b}$)

Factor, please:
${\displaystyle x^2+2x-24}$
a)${\displaystyle (x-6)(x+4)}$
b) ${\displaystyle (x-6)(x-4)}$
c) ${\displaystyle (x+6)(x+4)}$
d) ${\displaystyle (x+6)(x-4)}$